Calculus I · Limits and Continuity · exam

Limits and Continuity Practice Exam B

Take Limits and Continuity Practice Exam B as a realistic first-unit Calculus I assessment covering limits, asymptotes, continuity, and the IVT.

Course progressSupporting resourceReturn to the core path →

Practice Examination B

This exam is deliberately less patterned. It is meant to test whether you can identify methods rather than imitate section headings.

Suggested time: 90 minutes.

Exercise

Let

f(x)=\left\{\matrix{ x^2+1,\quad x<1,\quad 5,\quad x=1,\quad 3x-1,\quad x>1. }\right.

Find both one-sided limits, the two-sided limit, $f(1)$ , and determine continuity at $1$ .

Exercise

$\displaystyle\lim_{x\to2}\frac{x^3-8}{x^2-4}$

Exercise

$\displaystyle\lim_{x\to0}\frac{\sqrt{1+4x}-\sqrt{1-x}}x$

Exercise

$\displaystyle\lim_{x\to0}\frac{1-\cos(3x)}{x^2}$

Exercise

$\displaystyle\lim_{x\to0}\frac{\tan(2x)}{\sin(5x)}$

Exercise

$\displaystyle\lim_{x\to0}x\sin(1/x^2)$

Exercise

Find every discontinuity of $\dfrac{(x-1)(x+2)}{(x-1)(x-3)^2}$ , classify it, and state all one-sided infinite limits.

Exercise

$\displaystyle\lim_{x\to\infty}\frac{2x^3-x}{x^2+1}$

Exercise

Find the polynomial asymptote of the function in the previous problem.

Exercise

$\displaystyle\lim_{x\to-\infty}(\sqrt{x^2+10x}+x)$

Exercise

Find $a,b$ so

g(x)=\left\{\matrix{ ax+b,\quad x<1,\quad x^2+2,\quad 1\le x<2,\quad 3x+a,\quad x\ge2 }\right.

is continuous at $1$ and $2$ .

Exercise

Show that $x^5-3x+1=0$ has a root in $(0,1)$ , then perform two bisection steps.

Exercise

Prove $\lim_{x\to1}(4x-3)=1$ using the formal definition.

Exercise

A student claims that because $f(-1)<0<f(1)$ , every function has a root in $(-1,1)$ . Give a counterexample and state the missing hypothesis.

Full answers and selected worked solutions are in the appendices. Do not read them until you have completed an honest attempt. The universe already contains enough answer keys masquerading as education.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.

Calculus I · Limits and Continuity · exam

Limits and Continuity Practice Exam B

Take Limits and Continuity Practice Exam B as a realistic first-unit Calculus I assessment covering limits, asymptotes, continuity, and the IVT.

Course progressSupporting resourceReturn to the core path →

Practice Examination B

This exam is deliberately less patterned. It is meant to test whether you can identify methods rather than imitate section headings.

Suggested time: 90 minutes.

Exercise

Let

f(x)=\left\{\matrix{ x^2+1,\quad x<1,\quad 5,\quad x=1,\quad 3x-1,\quad x>1. }\right.

Find both one-sided limits, the two-sided limit, $f(1)$ , and determine continuity at $1$ .

Exercise

$\displaystyle\lim_{x\to2}\frac{x^3-8}{x^2-4}$

Exercise

$\displaystyle\lim_{x\to0}\frac{\sqrt{1+4x}-\sqrt{1-x}}x$

Exercise

$\displaystyle\lim_{x\to0}\frac{1-\cos(3x)}{x^2}$

Exercise

$\displaystyle\lim_{x\to0}\frac{\tan(2x)}{\sin(5x)}$

Exercise

$\displaystyle\lim_{x\to0}x\sin(1/x^2)$

Exercise

Find every discontinuity of $\dfrac{(x-1)(x+2)}{(x-1)(x-3)^2}$ , classify it, and state all one-sided infinite limits.

Exercise

$\displaystyle\lim_{x\to\infty}\frac{2x^3-x}{x^2+1}$

Exercise

Find the polynomial asymptote of the function in the previous problem.

Exercise

$\displaystyle\lim_{x\to-\infty}(\sqrt{x^2+10x}+x)$

Exercise

Find $a,b$ so

g(x)=\left\{\matrix{ ax+b,\quad x<1,\quad x^2+2,\quad 1\le x<2,\quad 3x+a,\quad x\ge2 }\right.

is continuous at $1$ and $2$ .

Exercise

Show that $x^5-3x+1=0$ has a root in $(0,1)$ , then perform two bisection steps.

Exercise

Prove $\lim_{x\to1}(4x-3)=1$ using the formal definition.

Exercise

A student claims that because $f(-1)<0<f(1)$ , every function has a root in $(-1,1)$ . Give a counterexample and state the missing hypothesis.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

Practice Examination B

Related resources

Original instruction with traceable references.

Practice Examination B

Related resources

Original instruction with traceable references.