Limits | Practice Problems, Examples & Solutions

Tangents

21st Century Astronomy

When viewed by radio telescopes, Jupiter is the second-brightest object in the sky. What is the source of its radiation?

Worlds of Gas and Liquid-The Giant Planets

03:50

Calculus: Early Transcendentals

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.

$$\mathbf{F}=\langle y,-x\rangle ; C=\left\{(x, y): x^{2}+y^{2}=1\right\}$$

Vector Calculus

Vector Fields

01:05

Calculus and Its Applications

(a) Draw the graph of any function $f(x)$ that passes through the point (3,2)
(b) Choose a point to the right of $x=3$ on the $x$ -axis and label it $3+h$
(c) Draw the straight line through the points $(3, f(3))$ and $(3+h, f(3+h))$
(d) What is the slope of this straight line (in terms of $h$ )?

The Derivative

The Slope of a Straight Line

Limits of a Function

871 Practice Problems

05:08

Calculus for Scientists and Engineers: Early Transcendental

Proving that $\lim _{x \rightarrow a} f(x) \neq L$ Use the following definition for the nonexistence of a limit. Assume $f$ is defined for all values of $x$ near a, except possibly at a. We say that $\lim _{x \rightarrow a} f(x) \neq L$ iffor some $\varepsilon>0$ there is no value of $\delta>0$ satisfying the condition $$ |f(x)-L|<\varepsilon \text { whenever } 0<|x-a|<\delta $$
Let
$$
f(x)=\left\{\begin{array}{ll}
0 & \text { if } x \text { is rational } \\
1 & \text { if } x \text { is irrational. }
\end{array}\right.
$$
Prove that $\lim _{x \rightarrow a} f(x)$ does not exist for any value of $a$. (Hint:
Assume $\left.\lim _{x \rightarrow a} f(x)=L \text { for some values of } a \text { and } L \text { and let } \varepsilon=\frac{1}{2} .\right)$

Limits

Precise Definitions of Limits

01:21

Calculus for Scientists and Engineers: Early Transcendental

Definition of a limit at infinity The limit at infinity $\lim _{x \rightarrow \infty} f(x)=L$ means that for any $\varepsilon>0,$ there exists $N>0$ such that $$ |f(x)-L|<\varepsilon \text { whenever } x>N $$
Use this definition to prove the following statements.
$$\lim _{x \rightarrow \infty} \frac{2 x+1}{x}=2$$

Limits

Precise Definitions of Limits

01:14

Calculus for Scientists and Engineers: Early Transcendental

Use the following definitions. Assume f exists for all $x$ near a with $x>$ a. We say that the limit of $f(x)$ as $x$ approaches a from the right of a is $L$ and write $\lim _{x \rightarrow a^{+}} f(x)=L,$ iffor any $\varepsilon>0$ there exists $\delta>0$ such that $$|f(x)-L|<\varepsilon \text { whenever } 0<x-a<\delta$$ Assume f exists for all values of $x$ near a with $x<a .$ We say that the limit of $f(x)$ as $x$ approaches a from the left of a is $L$ and write $\lim _{x \rightarrow a} f(x)=L,$ iffor any $\varepsilon>0$ there exists $\delta>0$ such that $$ |f(x)-L|<\varepsilon \text { whenever } 0<a-x<\delta $$
One-sided limit proof Prove that $\lim _{x \rightarrow 0^{+}} \sqrt{x}=0$

Limits

Precise Definitions of Limits

Continuity

197 Practice Problems

03:34

Calculus for Scientists and Engineers: Early Transcendental

Evaluate the following limits.
$$\lim _{(x, y, z) \rightarrow(1, \ln 2,3)} z e^{x y}$$

Functions of Several Variables

Limits and Continuity

06:43

Calculus for Scientists and Engineers: Early Transcendental

Continuity of $\sin x$ and $\cos x$
a. Use the identity $\sin (a+h)=\sin a \cos h+\cos a \sin h$ with the fact that $\lim _{x \rightarrow 0} \sin x=0$ to prove that $\lim \sin x=\sin a$ thereby establishing that $\sin x$ is continuous for all $x$. (Hint: Let $h=x-a \text { so that } x=a+h \text { and note that } h \rightarrow 0 \text { as } x \rightarrow a .)$
b. Use the identity $\cos (a+h)=\cos a \cos h-\sin a \sin h$ with the fact that $\lim _{x \rightarrow 0} \cos x=1$ to prove that $\lim _{x \rightarrow a} \cos x=\cos a$.

Limits

Continuity

00:53

Calculus for Scientists and Engineers: Early Transcendental

Violation of the Intermediate Value Theorem? Let $f(x)=\frac{|x|}{x} .$ Then $f(-2)=-1$ and $f(2)=1 .$ Therefore
$f(-2)<0<f(2),$ but there is no value of $c$ between -2 and 2 for which $f(c)=0 .$ Does this fact violate the Intermediate Value Theorem? Explain.

Limits

Continuity

Horizontal Asymptotes

141 Practice Problems

01:59

Calculus: Early Transcendental Functions

Sketch a graph of the function showing all extreme, intercepts and asymptotes.
$$f(x)=\frac{x-1}{x^{2}+4 x+3}$$

Preliminaries

Graphing Calculators and Computer Algebra Systems

00:58

Calculus: Early Transcendental Functions

Sketch a graph of the function showing all extreme, intercepts and asymptotes.
$$f(x)=3-x^{2}$$

Preliminaries

Graphing Calculators and Computer Algebra Systems

01:55

Calculus: Early Transcendentals

Horizontal and vertical asymptotes.
a. Analyze $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x),$ and then identify any horizontal asymptotes.
b. Find the vertical asymptotes. For each vertical asymptote $x=a$ analyze $\lim _{x \rightarrow a^{-}} f(x)$ and $\lim _{x \rightarrow a^{+}} f(x)$.
$$f(x)=\frac{x^{2}-4 x+3}{x-1}$$

Limits

Limits at Infinity

Limits using Graphs

334 Practice Problems

01:18

Calculus for Scientists and Engineers: Early Transcendental

Limits of composite functions
If $\lim _{x \rightarrow 1} f(x)=4,$ find $\lim _{x \rightarrow-1} f\left(x^{2}\right).$

Limits

Techniques for Computing Limits

02:26

Calculus for Scientists and Engineers: Early Transcendental

Evaluate the following limits.
$$\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x}-1}$$

Limits

Techniques for Computing Limits

01:56

Calculus for Scientists and Engineers: Early Transcendental

Calculate the following limits using the factorization formula $x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right),$ where $n$ is a positive integer and a is a real number.
$$\lim _{x \rightarrow-1} \frac{x^{7}+1}{x+1} \text { (Hint: Use the formula for $x^{7}-a^{7}$ with $a=-1 .$) }$$

Limits

Techniques for Computing Limits

Estimating Limits with Calculation

275 Practice Problems

00:44

Calculus for Scientists and Engineers: Early Transcendental

Evaluate the following limits, where c and $k$ are constants.
$$\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{2}{x^{2}-2 x}\right)$$

Limits

Techniques for Computing Limits

05:11

Calculus for Scientists and Engineers: Early Transcendental

Slope of a tangent line
a. Sketch a graph of $y=2^{x}$ and carefully draw three secant lines connecting the points $P(0,1)$ and $Q\left(x, 2^{x}\right),$ for $x=-3,-2$ and $-1.$
b. Find the slope of the line that joins $P(0,1)$ and $Q\left(x, 2^{x}\right),$ for $x \neq 0.$
c. Complete the table and make a conjecture about the value of $\lim _{x \rightarrow 0^{-}} \frac{2^{x}-1}{x}.$
$$\begin{array}{|c|c|c|c|c|c|c|}
\hline x & -1 & -0.1 & -0.01 & -0.001 & -0.0001 & -0.00001 \\
\hline \frac{2^{x}-1}{x} & & & & & \\
\hline
\end{array}$$

Limits

Techniques for Computing Limits

01:00

Calculus for Scientists and Engineers: Early Transcendental

Assume $\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3$ and $\lim _{x \rightarrow 1} h(x)=2 .$ Compute the following limits and state the limit laws used to justify your computations.
$$\lim _{x \rightarrow 1}\left[\frac{f(x)}{g(x)-h(x)}\right]$$

Limits

Techniques for Computing Limits

Left sided, right sided, vs two sided limits

128 Practice Problems

02:08

Calculus Early Transcendentals

Write out the definition for the corresponding limit in the marginal note on page $105,$ and use your definition to prove that the stated limit is correct.
(a) $\lim _{x \rightarrow+\infty}(x+1)=+\infty$
(b) $\lim _{x \rightarrow-\infty}(x+1)=-\infty$

LIMITS AND CONTINUITY

Limits (Discussed More Rigorously)

01:03

Calculus Early Transcendentals

Use the definitions in Exercise 27 to prove that the state one-sided limit is correct.
$$\lim _{x \rightarrow 2^{+}}(x+1)=3$$

LIMITS AND CONTINUITY

Limits (Discussed More Rigorously)

04:42

Calculus Early Transcendentals

Give rigorous definitions of $\lim _{x \rightarrow a^{+}} f(x)=L$ and $\lim _{x \rightarrow a^{-}} f(x)=L$.

LIMITS AND CONTINUITY

Limits (Discussed More Rigorously)

$\varepsilon-\delta$ definition of Limits

93 Practice Problems

02:21

Calculus: Early Transcendentals

Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between $\varepsilon$ and $\delta$ that guarantees the limit exists.
$\lim _{x \rightarrow a} b=b,$ for any constants $a$ and $b$

Limits

Precise Definitions of Limits

03:21

Calculus: Early Transcendentals

Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between $\varepsilon$ and $\delta$ that guarantees the limit exists.
$$\lim _{x \rightarrow 7} f(x)=9, \text { where } f(x)=\left\{\begin{array}{ll}
3 x-12 & \text { if } x \leq 7 \\
x+2 & \text { if } x>7
\end{array}\right.$$

Limits

Precise Definitions of Limits

07:15

Calculus: Early Transcendentals

Finding a symmetric interval The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=3 .$ For each value of $\varepsilon,$ find all values of $\delta>0$ such that
$$|f(x)-3|<\varepsilon \quad \text { whenever } \quad 0<|x-2|<\delta$$
a. $\varepsilon=1$
b. $\varepsilon=\frac{1}{2}$
c. For any $\varepsilon>0,$ make a conjecture about the corresponding values of $\delta$ satisfying (2)

Limits

Precise Definitions of Limits

Limit Rules

161 Practice Problems

01:43

Calculus Early Transcendentals

Use Definition 1.4.2 or 1.4.3 to prove that the stated limit is correct.
$$\lim _{x \rightarrow-\infty} \frac{4 x-1}{2 x+5}=2$$

LIMITS AND CONTINUITY

Limits (Discussed More Rigorously)

02:02

Calculus Early Transcendentals

Use Definition 1.4 .1 to prove that the limit is correct.
$$\lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}=-6$$

LIMITS AND CONTINUITY

Limits (Discussed More Rigorously)

01:35

Calculus Early Transcendentals

Use Definition 1.4 .1 to prove that the limit is correct.
$\lim _{x \rightarrow 2} 3=3$

LIMITS AND CONTINUITY

Limits (Discussed More Rigorously)

Limit Rules: Constant Multiple

8 Practice Problems

00:20

A Graphical Approach to Precalculus with Limits

Determine each limit, if it exists.
$$\lim _{x \rightarrow-3} 7$$

Limits, Derivatives, and Definite Integrals

Techniques for Calculating Limits

00:32

A Graphical Approach to Precalculus with Limits

Let $\lim _{x \rightarrow 4} f(x)=16$ and $\lim _{x \rightarrow 4} g(x)=8 .$ Use the limit rules to find each limit. Do not use a calculator.
$$\lim _{x \rightarrow 4}[f(x)-g(x)]$$

Limits, Derivatives, and Definite Integrals

Techniques for Calculating Limits

01:07

Precalculus 6th

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.
$$\lim _{x \rightarrow 2}(5 x-8)^{3}$$

Introduction to Calculus

Finding Limits Using Properties of Limits

Limit Rules: Sum/ Difference

48 Practice Problems

00:36

Calculus for Scientists and Engineers: Early Transcendental

Evaluate the following limits.
$$\lim _{x \rightarrow 2}\left(x^{2}-x\right)^{5}$$

Limits

Techniques for Computing Limits

00:12

Calculus for Scientists and Engineers: Early Transcendental

Evaluate the following limits.
$$\lim _{x \rightarrow 6} 4$$

Limits

Techniques for Computing Limits

02:29

Precalculus

Use a table of values to evaluate each limit. Compare each result with that of the corresponding Exercises in 15 through 24
$$\lim _{x \rightarrow-3}[a(x)+c(x)]$$

Bridges to Calculus: An Introduction to Limits

The Properties of Limits

Limit Rules: Multiplication/Quotient

50 Practice Problems

02:44

Precalculus

Evaluate the limits using limit properties. If a limit does not exist, state why.
$$\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$$

Bridges to Calculus: An Introduction to Limits

The Properties of Limits

01:17

Precalculus

Evaluate the limits using the limit properties.
$$\lim _{x \rightarrow-1} \frac{\left(\frac{x}{2 x+1}\right)^{2}-3 x}{\sqrt{x^{2}+3}+1}$$

Bridges to Calculus: An Introduction to Limits

The Properties of Limits

02:09

Precalculus

Evaluate the limits using the limit properties.
$$\lim _{x \rightarrow-3} \frac{x^{2}-2 x+3}{x-3}$$

Bridges to Calculus: An Introduction to Limits

The Properties of Limits

Infinite Limits

228 Practice Problems

00:11

Calculus for Scientists and Engineers: Early Transcendental

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$f(x)=\frac{1}{\sqrt{x} \sec x}$$

Limits

Infinite Limits

00:44

Calculus for Scientists and Engineers: Early Transcendental

Finding a function with infinite limits Give a formula for a function $f$ that satisfies $\lim _{x \rightarrow 6^{+}} f(x)=\infty$ and $\lim _{x \rightarrow 6^{-}} f(x)=-\infty$

Limits

Infinite Limits

00:58

Calculus for Scientists and Engineers: Early Transcendental

Trigonometric limits Investigate the following limits. $$\lim _{\theta \rightarrow \pi / 2^{+}} \frac{1}{3} \tan \theta$$

Limits

Infinite Limits

Infinite Limits: Horizontal vs Vertical Asymptotes

121 Practice Problems

05:15

Calculus for Scientists and Engineers: Early Transcendental

Limits of exponentials Evaluate $\lim _{T \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ Then state the horizontal asymptote(s) of $f$. Confirm your findings by plotting $f.$
$$f(x)=\frac{3 e^{x}+e^{-x}}{e^{x}+e^{-x}}$$

Limits

Limits at Infinity

00:57

Calculus for Scientists and Engineers: Early Transcendental

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence $\{2,4,6,8, \ldots\}$ is specified by the function $f(n)=2 n$ where $n=1,2,3, \ldots . .$ The limit of such a sequence is $\lim _{n \rightarrow \infty} f(n)$ provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist.
$\left\{\frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \ldots\right\},$ which is defined by $f(n)=\frac{n^{2}}{n+1}$ for $n=1,2,3, \ldots$

Limits

Limits at Infinity

00:53

Calculus for Scientists and Engineers: Early Transcendental

Steady states If a function $f$ represents a system that varies in time, the existence of $\lim _{t \rightarrow \infty} $f(t)$ means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value.
The amount of drug (in milligrams) in the blood after an IV tube is inserted is $m(t)=200\left(1-2^{-t}\right).$

Limits

Limits at Infinity

Squeeze (Sandwich) Theorem

17 Practice Problems

02:05

Calculus: Early Transcendental Functions

Use the Squeeze Theorem to prove that $\lim _{x \rightarrow 0^{+}}\left[\sqrt{x} \cos ^{2}(1 / x)\right]=0$
Identify the functions $f$ and $h,$ show graphically that $f(x) \leq \sqrt{x} \cos ^{2}(1 / x) \leq h(x)$ for all $x > 0,$ and justify
$\lim _{x \rightarrow 0^{+}} f(x)=0$ and $\lim _{x \rightarrow 0^{+}} h(x)=0$

Limits and Continuity

Computation of Limits

03:13

Thomas Calculus

The Sandwich Theorem for functions of two variables states that if $g(x, y) \leq f(x, y) \leq h(x, y)$ for all $(x, y) \neq\left(x_{0}, y_{0}\right)$ in a disk centered at $\left(x_{0}, y_{0}\right)$ and if $g$ and $h$ have the same finite limit $L$ as $(x, y) \rightarrow\left(x_{0}, y_{0}\right),$ then
$$
\lim _{(x, y) \rightarrow\left(x_{0}, y_{0}\right)} f(x, y)=L
$$
Use this result to support your answers to the questions in Exercises $45-48 .$
Does knowing that
$$
2|x y|-\frac{x^{2} y^{2}}{6}<4-4 \cos \sqrt{|x y|}<2|x y|
$$
tell you anything about
$$
\lim _{(x, y) \rightarrow(0,0)} \frac{4-4 \cos \sqrt{|x y|}}{|x y|} ?
$$
Give reasons for your answer.

Partial Derivatives

Limits and Continuity in Higher Dimensions

00:51

Calculus Volume 1

In the following exercises, use the squeeze theorem to prove the limit.
$$\lim _{x \rightarrow 0} x^{2} \cos (2 \pi x)=0$$

Limits

The Precise Definition of a Limit