Calculus I · Limits and Continuity · lesson

Limit Laws and How to Use Them

Learn Limit Laws and How to Use Them with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.

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Where this chapter fits

Chapter 2: Finite limits and algebra

Turn indeterminate forms into solvable expressions using substitution, limit laws, factoring, conjugates, and piecewise reasoning.

Reading lens: What did direct substitution reveal, and which algebraic move removes the obstacle without changing nearby behavior? Keep that question in view while reading Limit Laws and How to Use Them; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects Direct Substitution for Limits to What 0/0 Means in a Limit. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Use sums, differences, constant multiples, products, quotients, powers, and roots to build limits from simpler limits.

Limit Laws

Suppose

\lim_{x\to a}f(x)=L \qquad\text{and}\qquad \lim_{x\to a}g(x)=M.

The basic limit laws are:

Theorem

Limit Laws

\lim_{x\to a}[f(x)+g(x)]\quad =L+M,\quad \lim_{x\to a}[f(x)-g(x)]\quad =L-M,\quad \lim_{x\to a}[cf(x)]\quad =cL,\quad \lim_{x\to a}[f(x)g(x)]\quad =LM,\quad \lim_{x\to a}\frac{f(x)}{g(x)}\quad =\frac{L}{M},\qquad M\ne0,\quad \lim_{x\to a}[f(x)]^n\quad =L^n.

When the relevant root is defined,

\lim_{x\to a}\sqrt[n]{f(x)}=\sqrt[n]{L}.

Concept

The laws say that limits cooperate with ordinary arithmetic, provided the arithmetic itself remains legal. If one part approaches $3$ and another approaches $5$ , their sum approaches $8$ . The only major warning in the basic laws is division: the denominator's limit may not be zero.

Guided walkthrough

Building a limit from known pieces

Suppose

\lim_{x\to2}f(x)=3 \qquad\text{and}\qquad \lim_{x\to2}g(x)=5.

Find

\lim_{x\to2}[2f(x)+g(x)].

Show worked solution

The constant multiple law gives

\lim_{x\to2}2f(x)=2(3)=6.

Then the sum law gives

6+5=11.

Therefore,

\boxed{\lim_{x\to2}[2f(x)+g(x)]=11}.

Worked example

Several laws at once

Suppose

\lim_{x\to1}f(x)=-2, \qquad \lim_{x\to1}g(x)=4.

Evaluate

\lim_{x\to1}\frac{[f(x)]^2+3g(x)}{g(x)-f(x)}.

Show worked solution

Apply the laws to each part:

[f(x)]^2\to(-2)^2=4,

3g(x)\to3(4)=12,

so the numerator approaches

4+12=16.

The denominator approaches

4-(-2)=6.

Because $6\ne0$ , the quotient law applies:

\boxed{\frac{16}{6}=\frac83}.

Quick check

If $f(x)\to3$ and $g(x)\to-1$ as $x\to4$ , evaluate

\lim_{x\to4}[f(x)g(x)+2f(x)].

Answer. $3(-1)+2(3)=3$ .

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.

Chapter 2: Finite limits and algebra

Turn indeterminate forms into solvable expressions using substitution, limit laws, factoring, conjugates, and piecewise reasoning.

This page connects Direct Substitution for Limits to What 0/0 Means in a Limit. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Limit Laws

Suppose

\lim_{x\to a}f(x)=L \qquad\text{and}\qquad \lim_{x\to a}g(x)=M.

The basic limit laws are:

Theorem

Limit Laws

\lim_{x\to a}[f(x)+g(x)]\quad =L+M,\quad \lim_{x\to a}[f(x)-g(x)]\quad =L-M,\quad \lim_{x\to a}[cf(x)]\quad =cL,\quad \lim_{x\to a}[f(x)g(x)]\quad =LM,\quad \lim_{x\to a}\frac{f(x)}{g(x)}\quad =\frac{L}{M},\qquad M\ne0,\quad \lim_{x\to a}[f(x)]^n\quad =L^n.

When the relevant root is defined,

\lim_{x\to a}\sqrt[n]{f(x)}=\sqrt[n]{L}.

Concept

Guided walkthrough

Building a limit from known pieces

Suppose

\lim_{x\to2}f(x)=3 \qquad\text{and}\qquad \lim_{x\to2}g(x)=5.

Find

\lim_{x\to2}[2f(x)+g(x)].

Show worked solution

The constant multiple law gives

\lim_{x\to2}2f(x)=2(3)=6.

Then the sum law gives

6+5=11.

Therefore,

\boxed{\lim_{x\to2}[2f(x)+g(x)]=11}.

Worked example

Several laws at once

Suppose

\lim_{x\to1}f(x)=-2, \qquad \lim_{x\to1}g(x)=4.

Evaluate

\lim_{x\to1}\frac{[f(x)]^2+3g(x)}{g(x)-f(x)}.

Show worked solution

Apply the laws to each part:

[f(x)]^2\to(-2)^2=4,

3g(x)\to3(4)=12,

so the numerator approaches

4+12=16.

The denominator approaches

4-(-2)=6.

Because $6\ne0$ , the quotient law applies:

\boxed{\frac{16}{6}=\frac83}.

Quick check

If $f(x)\to3$ and $g(x)\to-1$ as $x\to4$ , evaluate

\lim_{x\to4}[f(x)g(x)+2f(x)].

Answer. $3(-1)+2(3)=3$ .

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.

Chapter 2: Finite limits and algebra

Limit Laws

Related resources

Original instruction with traceable references.

Chapter 2: Finite limits and algebra

Limit Laws

Related resources

Original instruction with traceable references.