Worksheet · Calculus I · Unit 1

Evaluating Limits Worksheet with Complete Solutions

Twenty-four limits that progress from substitution to algebraic, one-sided, and infinite behavior.

24 problems55 min estimated timeFoundational to intermediate progression

What is included

Practice evaluating limits with a printable student worksheet, answer key, and accessible worked solutions.

Skills assessed

direct substitution
factoring
rationalization
one-sided limits
limits at infinity

Prerequisites

function notation
algebraic factoring

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF

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Evaluate $lim_{x\to2}(3x^2-5x+4)$ .
Evaluate $lim_{x\to-2}(x^3+4x)$ .
Evaluate $lim_{t\to3}\frac{t^2+1}{t+2}$ .
Evaluate $lim_{h\to0}(7-4h+h^2)$ .
Evaluate $lim_{u\to1}\sqrt{u+8}$ .
Evaluate $lim_{x\to4}\frac{x+5}{\sqrt{x}}$ .
Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .
Evaluate $\lim_{x\to-4}\frac{x^2-16}{x+4}$ .
Evaluate $\lim_{x\to2}\frac{x^3-8}{x-2}$ .
Evaluate $\lim_{x\to1}\frac{x^2+x-2}{x-1}$ .
Evaluate $\lim_{y\to5}\frac{y^2-7y+10}{y-5}$ .
Evaluate $\lim_{z\to-1}\frac{z^3+1}{z+1}$ .
Evaluate $\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$ .
Evaluate $\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$ .
Evaluate $\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}$ .
Evaluate $\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}$ .
For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .
For $f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}$ , evaluate the limit as $x\to0^+$ .
For $g(x)=\frac{|x|}{x}$ , evaluate the limit as $x\to0^-$ .
For $h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}$ , evaluate the limit as $x\to2$ .
Evaluate $\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}$ .
Evaluate $\lim_{x\to\infty}\frac{3x+7}{x^2+1}$ .
Evaluate $\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}$ .
Evaluate $\lim_{x\to2^+}\frac{1}{x-2}$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Evaluate $lim_{x\to2}(3x^2-5x+4)$ .

Answer: $6$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $1$ .
Substitute the target and simplify to obtain $6$ .

Problem 2: Evaluate $lim_{x\to-2}(x^3+4x)$ .

Answer: $-16$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $-2$ .
Substitute the target and simplify to obtain $-16$ .

Problem 3: Evaluate $lim_{t\to3}\frac{t^2+1}{t+2}$ .

Answer: $2$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $3$ .
Substitute the target and simplify to obtain $2$ .

Problem 4: Evaluate $lim_{h\to0}(7-4h+h^2)$ .

Answer: $7$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $0$ .
Substitute the target and simplify to obtain $7$ .

Problem 5: Evaluate $lim_{u\to1}\sqrt{u+8}$ .

Answer: $3$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $1$ .
Substitute the target and simplify to obtain $3$ .

Problem 6: Evaluate $lim_{x\to4}\frac{x+5}{\sqrt{x}}$ .

Answer: $\frac92$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $4$ .
Substitute the target and simplify to obtain $\frac92$ .

Problem 7: Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .

Answer: $6$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x-3)(x+3)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $6$ .

Problem 8: Evaluate $\lim_{x\to-4}\frac{x^2-16}{x+4}$ .

Answer: $-8$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x+4)(x-4)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $-8$ .

Problem 9: Evaluate $\lim_{x\to2}\frac{x^3-8}{x-2}$ .

Answer: $12$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x-2)(x^2+2x+4)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $12$ .

Problem 10: Evaluate $\lim_{x\to1}\frac{x^2+x-2}{x-1}$ .

Answer: $3$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x-1)(x+2)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $3$ .

Problem 11: Evaluate $\lim_{y\to5}\frac{y^2-7y+10}{y-5}$ .

Answer: $3$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(y-5)(y-2)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $3$ .

Problem 12: Evaluate $\lim_{z\to-1}\frac{z^3+1}{z+1}$ .

Answer: $3$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(z+1)(z^2-z+1)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $3$ .

Problem 13: Evaluate $\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$ .

Answer: $\frac16$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{x}+3$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $\frac16$ .

Problem 14: Evaluate $\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$ .

Answer: $\frac12$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{1+x}+1$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $\frac12$ .

Problem 15: Evaluate $\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}$ .

Answer: $4$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{t}+2$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $4$ .

Problem 16: Evaluate $\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}$ .

Answer: $6$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{9+u}+3$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $6$ .

Problem 17: For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .

Answer: $3$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $3$ .

Problem 18: For $f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}$ , evaluate the limit as $x\to0^+$ .

Answer: $1$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $1$ .

Problem 19: For $g(x)=\frac{|x|}{x}$ , evaluate the limit as $x\to0^-$ .

Answer: $-1$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $-1$ .

Problem 20: For $h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}$ , evaluate the limit as $x\to2$ .

Answer: $4$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $4$ .

Problem 21: Evaluate $\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}$ .

Answer: $\frac52$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: equal degrees.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $\frac52$ .

Problem 22: Evaluate $\lim_{x\to\infty}\frac{3x+7}{x^2+1}$ .

Answer: $0$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: denominator degree is larger.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $0$ .

Problem 23: Evaluate $\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}$ .

Answer: $2$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: equal degrees.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $2$ .

Problem 24: Evaluate $\lim_{x\to2^+}\frac{1}{x-2}$ .

Answer: $+\infty$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: positive denominator approaching zero.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $+\infty$ .

Common errors

Substituting before checking whether the form is determinate.
Cancelling terms instead of common factors.
Ignoring approach direction at jumps.

Worksheet · Calculus I · Unit 1

Evaluating Limits Worksheet with Complete Solutions

Twenty-four limits that progress from substitution to algebraic, one-sided, and infinite behavior.

24 problems55 min estimated timeFoundational to intermediate progression

What is included

Practice evaluating limits with a printable student worksheet, answer key, and accessible worked solutions.

Skills assessed

direct substitution
factoring
rationalization
one-sided limits
limits at infinity

Prerequisites

function notation
algebraic factoring

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF

Device-only study control

Ready for an honest first attempt?

This control sends only the resource id and type—never your answers or identity.

Printable preview

Evaluate $lim_{x\to2}(3x^2-5x+4)$ .
Evaluate $lim_{x\to-2}(x^3+4x)$ .
Evaluate $lim_{t\to3}\frac{t^2+1}{t+2}$ .
Evaluate $lim_{h\to0}(7-4h+h^2)$ .
Evaluate $lim_{u\to1}\sqrt{u+8}$ .
Evaluate $lim_{x\to4}\frac{x+5}{\sqrt{x}}$ .
Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .
Evaluate $\lim_{x\to-4}\frac{x^2-16}{x+4}$ .
Evaluate $\lim_{x\to2}\frac{x^3-8}{x-2}$ .
Evaluate $\lim_{x\to1}\frac{x^2+x-2}{x-1}$ .
Evaluate $\lim_{y\to5}\frac{y^2-7y+10}{y-5}$ .
Evaluate $\lim_{z\to-1}\frac{z^3+1}{z+1}$ .
Evaluate $\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$ .
Evaluate $\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$ .
Evaluate $\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}$ .
Evaluate $\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}$ .
For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .
For $f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}$ , evaluate the limit as $x\to0^+$ .
For $g(x)=\frac{|x|}{x}$ , evaluate the limit as $x\to0^-$ .
For $h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}$ , evaluate the limit as $x\to2$ .
Evaluate $\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}$ .
Evaluate $\lim_{x\to\infty}\frac{3x+7}{x^2+1}$ .
Evaluate $\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}$ .
Evaluate $\lim_{x\to2^+}\frac{1}{x-2}$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Evaluate $lim_{x\to2}(3x^2-5x+4)$ .

Answer: $6$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $1$ .
Substitute the target and simplify to obtain $6$ .

Problem 2: Evaluate $lim_{x\to-2}(x^3+4x)$ .

Answer: $-16$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $-2$ .
Substitute the target and simplify to obtain $-16$ .

Problem 3: Evaluate $lim_{t\to3}\frac{t^2+1}{t+2}$ .

Answer: $2$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $3$ .
Substitute the target and simplify to obtain $2$ .

Problem 4: Evaluate $lim_{h\to0}(7-4h+h^2)$ .

Answer: $7$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $0$ .
Substitute the target and simplify to obtain $7$ .

Problem 5: Evaluate $lim_{u\to1}\sqrt{u+8}$ .

Answer: $3$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $1$ .
Substitute the target and simplify to obtain $3$ .

Problem 6: Evaluate $lim_{x\to4}\frac{x+5}{\sqrt{x}}$ .

Answer: $\frac92$

Why this method: Direct substitution matches the mathematical structure before any algebraic cleanup.

The expression is continuous at the target input $4$ .
Substitute the target and simplify to obtain $\frac92$ .

Problem 7: Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .

Answer: $6$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x-3)(x+3)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $6$ .

Problem 8: Evaluate $\lim_{x\to-4}\frac{x^2-16}{x+4}$ .

Answer: $-8$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x+4)(x-4)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $-8$ .

Problem 9: Evaluate $\lim_{x\to2}\frac{x^3-8}{x-2}$ .

Answer: $12$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x-2)(x^2+2x+4)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $12$ .

Problem 10: Evaluate $\lim_{x\to1}\frac{x^2+x-2}{x-1}$ .

Answer: $3$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(x-1)(x+2)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $3$ .

Problem 11: Evaluate $\lim_{y\to5}\frac{y^2-7y+10}{y-5}$ .

Answer: $3$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(y-5)(y-2)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $3$ .

Problem 12: Evaluate $\lim_{z\to-1}\frac{z^3+1}{z+1}$ .

Answer: $3$

Why this method: Factoring and cancellation matches the mathematical structure before any algebraic cleanup.

Factor the vanishing numerator as $(z+1)(z^2-z+1)$ .
Cancel the common factor only for nearby inputs, then substitute the target.
The simplified expression approaches $3$ .

Problem 13: Evaluate $\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$ .

Answer: $\frac16$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{x}+3$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $\frac16$ .

Problem 14: Evaluate $\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$ .

Answer: $\frac12$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{1+x}+1$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $\frac12$ .

Problem 15: Evaluate $\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}$ .

Answer: $4$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{t}+2$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $4$ .

Problem 16: Evaluate $\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}$ .

Answer: $6$

Why this method: Rationalization matches the mathematical structure before any algebraic cleanup.

Multiply numerator and denominator by the conjugate $\sqrt{9+u}+3$ .
Use the difference-of-squares identity and cancel the factor that tends to zero.
Substitution in the simplified expression gives $6$ .

Problem 17: For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .

Answer: $3$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $3$ .

Problem 18: For $f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}$ , evaluate the limit as $x\to0^+$ .

Answer: $1$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $1$ .

Problem 19: For $g(x)=\frac{|x|}{x}$ , evaluate the limit as $x\to0^-$ .

Answer: $-1$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $-1$ .

Problem 20: For $h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}$ , evaluate the limit as $x\to2$ .

Answer: $4$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Select the branch determined by the direction of approach.
Evaluate nearby behavior on that branch; the value assigned at the endpoint does not control the limit.
The required approach gives $4$ .

Problem 21: Evaluate $\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}$ .

Answer: $\frac52$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: equal degrees.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $\frac52$ .

Problem 22: Evaluate $\lim_{x\to\infty}\frac{3x+7}{x^2+1}$ .

Answer: $0$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: denominator degree is larger.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $0$ .

Problem 23: Evaluate $\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}$ .

Answer: $2$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: equal degrees.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $2$ .

Problem 24: Evaluate $\lim_{x\to2^+}\frac{1}{x-2}$ .

Answer: $+\infty$

Why this method: Dominant-term or sign analysis matches the mathematical structure before any algebraic cleanup.

Identify the dominant behavior: positive denominator approaching zero.
Divide by the highest relevant power or use the sign of the vanishing factor.
The limit is $+\infty$ .

Common errors

Substituting before checking whether the form is determinate.
Cancelling terms instead of common factors.
Ignoring approach direction at jumps.

What is included

Skills assessed

Prerequisites

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Evaluate limx→2(3x2−5x+4)lim_{x\to2}(3x^2-5x+4)limx→2​(3x2−5x+4).

Problem 2: Evaluate limx→−2(x3+4x)lim_{x\to-2}(x^3+4x)limx→−2​(x3+4x).

Problem 3: Evaluate limt→3t2+1t+2lim_{t\to3}\frac{t^2+1}{t+2}limt→3​t+2t2+1​.

Problem 4: Evaluate limh→0(7−4h+h2)lim_{h\to0}(7-4h+h^2)limh→0​(7−4h+h2).

Problem 5: Evaluate limu→1u+8lim_{u\to1}\sqrt{u+8}limu→1​u+8​.

Problem 6: Evaluate limx→4x+5xlim_{x\to4}\frac{x+5}{\sqrt{x}}limx→4​x​x+5​.

Problem 7: Evaluate lim⁡x→3x2−9x−3\lim_{x\to3}\frac{x^2-9}{x-3}limx→3​x−3x2−9​.

Problem 8: Evaluate lim⁡x→−4x2−16x+4\lim_{x\to-4}\frac{x^2-16}{x+4}limx→−4​x+4x2−16​.

Problem 9: Evaluate lim⁡x→2x3−8x−2\lim_{x\to2}\frac{x^3-8}{x-2}limx→2​x−2x3−8​.

Problem 10: Evaluate lim⁡x→1x2+x−2x−1\lim_{x\to1}\frac{x^2+x-2}{x-1}limx→1​x−1x2+x−2​.

Problem 11: Evaluate lim⁡y→5y2−7y+10y−5\lim_{y\to5}\frac{y^2-7y+10}{y-5}limy→5​y−5y2−7y+10​.

Problem 12: Evaluate lim⁡z→−1z3+1z+1\lim_{z\to-1}\frac{z^3+1}{z+1}limz→−1​z+1z3+1​.

Problem 13: Evaluate lim⁡x→9x−3x−9\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}limx→9​x−9x​−3​.

Problem 14: Evaluate lim⁡x→01+x−1x\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}limx→0​x1+x​−1​.

Problem 15: Evaluate lim⁡t→4t−4t−2\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}limt→4​t​−2t−4​.

Problem 16: Evaluate lim⁡u→0u9+u−3\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}limu→0​9+u​−3u​.

Problem 17: For f(x)={x+2,x<14−x,x≥1f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}f(x)={x+2,4−x,​x<1x≥1​, evaluate the limit as x→1−x\to1^-x→1−.

Problem 18: For f(x)={2x,x<0x2+1,x≥0f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}f(x)={2x,x2+1,​x<0x≥0​, evaluate the limit as x→0+x\to0^+x→0+.

Problem 19: For g(x)=∣x∣xg(x)=\frac{|x|}{x}g(x)=x∣x∣​, evaluate the limit as x→0−x\to0^-x→0−.

Problem 20: For h(x)={x2,x≤26−x,x>2h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}h(x)={x2,6−x,​x≤2x>2​, evaluate the limit as x→2x\to2x→2.

Problem 21: Evaluate lim⁡x→∞5x2−12x2+3\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}limx→∞​2x2+35x2−1​.

Problem 22: Evaluate lim⁡x→∞3x+7x2+1\lim_{x\to\infty}\frac{3x+7}{x^2+1}limx→∞​x2+13x+7​.

Problem 23: Evaluate lim⁡x→−∞4x3+x2x3−5\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}limx→−∞​2x3−54x3+x​.

Problem 24: Evaluate lim⁡x→2+1x−2\lim_{x\to2^+}\frac{1}{x-2}limx→2+​x−21​.

Common errors

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Prerequisites

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Complete worked solutions

Problem 1: Evaluate limx→2(3x2−5x+4)lim_{x\to2}(3x^2-5x+4)limx→2​(3x2−5x+4).

Problem 2: Evaluate limx→−2(x3+4x)lim_{x\to-2}(x^3+4x)limx→−2​(x3+4x).

Problem 3: Evaluate limt→3t2+1t+2lim_{t\to3}\frac{t^2+1}{t+2}limt→3​t+2t2+1​.

Problem 4: Evaluate limh→0(7−4h+h2)lim_{h\to0}(7-4h+h^2)limh→0​(7−4h+h2).

Problem 5: Evaluate limu→1u+8lim_{u\to1}\sqrt{u+8}limu→1​u+8​.

Problem 6: Evaluate limx→4x+5xlim_{x\to4}\frac{x+5}{\sqrt{x}}limx→4​x​x+5​.

Problem 7: Evaluate lim⁡x→3x2−9x−3\lim_{x\to3}\frac{x^2-9}{x-3}limx→3​x−3x2−9​.

Problem 8: Evaluate lim⁡x→−4x2−16x+4\lim_{x\to-4}\frac{x^2-16}{x+4}limx→−4​x+4x2−16​.

Problem 9: Evaluate lim⁡x→2x3−8x−2\lim_{x\to2}\frac{x^3-8}{x-2}limx→2​x−2x3−8​.

Problem 10: Evaluate lim⁡x→1x2+x−2x−1\lim_{x\to1}\frac{x^2+x-2}{x-1}limx→1​x−1x2+x−2​.

Problem 11: Evaluate lim⁡y→5y2−7y+10y−5\lim_{y\to5}\frac{y^2-7y+10}{y-5}limy→5​y−5y2−7y+10​.

Problem 12: Evaluate lim⁡z→−1z3+1z+1\lim_{z\to-1}\frac{z^3+1}{z+1}limz→−1​z+1z3+1​.

Problem 13: Evaluate lim⁡x→9x−3x−9\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}limx→9​x−9x​−3​.

Problem 14: Evaluate lim⁡x→01+x−1x\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}limx→0​x1+x​−1​.

Problem 15: Evaluate lim⁡t→4t−4t−2\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}limt→4​t​−2t−4​.

Problem 16: Evaluate lim⁡u→0u9+u−3\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}limu→0​9+u​−3u​.

Problem 17: For f(x)={x+2,x<14−x,x≥1f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}f(x)={x+2,4−x,​x<1x≥1​, evaluate the limit as x→1−x\to1^-x→1−.

Problem 18: For f(x)={2x,x<0x2+1,x≥0f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}f(x)={2x,x2+1,​x<0x≥0​, evaluate the limit as x→0+x\to0^+x→0+.

Problem 19: For g(x)=∣x∣xg(x)=\frac{|x|}{x}g(x)=x∣x∣​, evaluate the limit as x→0−x\to0^-x→0−.

Problem 20: For h(x)={x2,x≤26−x,x>2h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}h(x)={x2,6−x,​x≤2x>2​, evaluate the limit as x→2x\to2x→2.

Problem 21: Evaluate lim⁡x→∞5x2−12x2+3\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}limx→∞​2x2+35x2−1​.

Problem 22: Evaluate lim⁡x→∞3x+7x2+1\lim_{x\to\infty}\frac{3x+7}{x^2+1}limx→∞​x2+13x+7​.

Problem 23: Evaluate lim⁡x→−∞4x3+x2x3−5\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}limx→−∞​2x3−54x3+x​.

Problem 24: Evaluate lim⁡x→2+1x−2\lim_{x\to2^+}\frac{1}{x-2}limx→2+​x−21​.

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Problem 1: Evaluate $lim_{x\to2}(3x^2-5x+4)$ .

Problem 2: Evaluate $lim_{x\to-2}(x^3+4x)$ .

Problem 3: Evaluate $lim_{t\to3}\frac{t^2+1}{t+2}$ .

Problem 4: Evaluate $lim_{h\to0}(7-4h+h^2)$ .

Problem 5: Evaluate $lim_{u\to1}\sqrt{u+8}$ .

Problem 6: Evaluate $lim_{x\to4}\frac{x+5}{\sqrt{x}}$ .

Problem 7: Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .

Problem 8: Evaluate $\lim_{x\to-4}\frac{x^2-16}{x+4}$ .

Problem 9: Evaluate $\lim_{x\to2}\frac{x^3-8}{x-2}$ .

Problem 10: Evaluate $\lim_{x\to1}\frac{x^2+x-2}{x-1}$ .

Problem 11: Evaluate $\lim_{y\to5}\frac{y^2-7y+10}{y-5}$ .

Problem 12: Evaluate $\lim_{z\to-1}\frac{z^3+1}{z+1}$ .

Problem 13: Evaluate $\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$ .

Problem 14: Evaluate $\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$ .

Problem 15: Evaluate $\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}$ .

Problem 16: Evaluate $\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}$ .

Problem 17: For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .

Problem 18: For $f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}$ , evaluate the limit as $x\to0^+$ .

Problem 19: For $g(x)=\frac{|x|}{x}$ , evaluate the limit as $x\to0^-$ .

Problem 20: For $h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}$ , evaluate the limit as $x\to2$ .

Problem 21: Evaluate $\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}$ .

Problem 22: Evaluate $\lim_{x\to\infty}\frac{3x+7}{x^2+1}$ .

Problem 23: Evaluate $\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}$ .

Problem 24: Evaluate $\lim_{x\to2^+}\frac{1}{x-2}$ .

Problem 1: Evaluate $lim_{x\to2}(3x^2-5x+4)$ .

Problem 2: Evaluate $lim_{x\to-2}(x^3+4x)$ .

Problem 3: Evaluate $lim_{t\to3}\frac{t^2+1}{t+2}$ .

Problem 4: Evaluate $lim_{h\to0}(7-4h+h^2)$ .

Problem 5: Evaluate $lim_{u\to1}\sqrt{u+8}$ .

Problem 6: Evaluate $lim_{x\to4}\frac{x+5}{\sqrt{x}}$ .

Problem 7: Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .

Problem 8: Evaluate $\lim_{x\to-4}\frac{x^2-16}{x+4}$ .

Problem 9: Evaluate $\lim_{x\to2}\frac{x^3-8}{x-2}$ .

Problem 10: Evaluate $\lim_{x\to1}\frac{x^2+x-2}{x-1}$ .

Problem 11: Evaluate $\lim_{y\to5}\frac{y^2-7y+10}{y-5}$ .

Problem 12: Evaluate $\lim_{z\to-1}\frac{z^3+1}{z+1}$ .

Problem 13: Evaluate $\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}$ .

Problem 14: Evaluate $\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$ .

Problem 15: Evaluate $\lim_{t\to4}\frac{t-4}{\sqrt{t}-2}$ .

Problem 16: Evaluate $\lim_{u\to0}\frac{u}{\sqrt{9+u}-3}$ .

Problem 17: For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .

Problem 18: For $f(x)=\begin{cases}2x,&x<0\\x^2+1,&x\ge0\end{cases}$ , evaluate the limit as $x\to0^+$ .

Problem 19: For $g(x)=\frac{|x|}{x}$ , evaluate the limit as $x\to0^-$ .

Problem 20: For $h(x)=\begin{cases}x^2,&x\le2\\6-x,&x>2\end{cases}$ , evaluate the limit as $x\to2$ .

Problem 21: Evaluate $\lim_{x\to\infty}\frac{5x^2-1}{2x^2+3}$ .

Problem 22: Evaluate $\lim_{x\to\infty}\frac{3x+7}{x^2+1}$ .

Problem 23: Evaluate $\lim_{x\to-\infty}\frac{4x^3+x}{2x^3-5}$ .

Problem 24: Evaluate $\lim_{x\to2^+}\frac{1}{x-2}$ .