Continuous functions | Practice Problems, Examples & Solutions

Types of Discontinuities

Calculus: Early Transcendental Functions

Find all discontinuities of $f(x) .$ For each discontinuity that is removable, define a new function that removes the discontinuity.
$$f(x)=\left\{\begin{array}{ll}
2 x & \text { if } x<1 \\
x^{2} & \text { if } x \geq 1
\end{array}\right.$$

Limits and Continuity

Continuity and its Consequences

00:34

Calculus: Early Transcendental Functions

Find all discontinuities of $f(x) .$ For each discontinuity that is removable, define a new function that removes the discontinuity.
$$f(x)=\frac{x-1}{x^{2}-1}$$

Limits and Continuity

Continuity and its Consequences

00:32

Calculus: Early Transcendental Functions

Use the given graph to identify all discontinuities of the functions.
(FIGURE CAN'T COPY)

Limits and Continuity

Continuity and its Consequences

Types of Discontinuities: Holes

0 Practice Problems

Types of Discontinuities: Jump

0 Practice Problems

Types of Discontinuities: Vertical Asymptotes

39 Practice Problems

00:21

Calculus: Early Transcendental Functions

Adjust the graphing window to identify all vertical asympotes.
$$f(x)=\frac{x^{2}-1}{\sqrt{x^{4}+x}}$$

Preliminaries

Graphing Calculators and Computer Algebra Systems

01:18

Calculus: Early Transcendental Functions

Find all vertical asymptotes.
$$f(x)=\frac{x^{2}+1}{x^{3}+3 x^{2}+2 x}$$

Preliminaries

Graphing Calculators and Computer Algebra Systems

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

Limit Definition of Continuity

111 Practice Problems

00:59

Calculus: Early Transcendental Functions

A function is continuous from the right at $x=a$ if $\lim _{x \rightarrow a^{+}} f(x)=f(a) .$Determine whether $f(x)$ is continuous from the right at $x=2.$
$$f(x)=\left\{\begin{array}{ll}
x^{2} & \text { if } x<2 \\
3 x-1 & \text { if } x \geq 2
\end{array}\right.$$

Limits and Continuity

Continuity and its Consequences

01:39

Calculus: Early Transcendentals

Classify the discontinuities in the following functions at the given points.
$$f(x)=\frac{x^{2}-7 x+10}{x-2} ; x=2$$

Limits

Continuity

01:51

Calculus: Early Transcendentals

Functions with roots Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is $f$ continuous from the left or continuous from the right?
$$f(x)=\sqrt[3]{x^{2}-2 x-3}$$

Limits

Continuity

$\varepsilon-\delta$ Definition of Continuity

21 Practice Problems

00:29

Calculus and Its Applications

Is the function given by $g(x)=4 x^{3}-6 x$ continuous on $\mathbb{R} ?$

Differentiation

Algebraic limits and Continuity

01:29

Precalculus 6th

Determine for what numbers, if any, the given function is discontinuous.
$$f(x)=\frac{x+1}{(x+1)(x-4)}$$

Introduction to Calculus

Limits and Continuity

01:42

Precalculus 6th

Use the definition of continuity to determine whether $f$ is continuous at a.
$$\begin{aligned}&f(x)=\frac{x^{2}+8 x}{x^{2}-8 x}\\&a=0\end{aligned}$$

Introduction to Calculus

Limits and Continuity

Intermediate Value Theorem

36 Practice Problems

02:36

Calculus: Early Transcendentals

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

05:41

Calculus: Early Transcendentals

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval.
b. Use a graphing utility to find all the solutions to the equation on the given interval.
c. Illustrate your answers with an appropriate graph.
$$\sqrt{x^{4}+25 x^{3}+10}=5 ;(0,1)$$

Limits

Continuity

01:38

216

Use the intermediate value theorem to show that Fhas a zero between $a$ and $b$.
$$f(x)=2 x^{4}+3 x-2 ; \quad a=0, \quad b=1$$

Polynomial and Rational Function

Polynomial Functions of Degree Greater Than

Composition Rule for Limits

3 Practice Problems

00:43

Precalculus

For Exercises 61 through $64,$ evaluate the limits by dividing the numerator and denominator by the highest power of $x$ occurring in the denominator.
$$\lim _{x \rightarrow \infty} \frac{3 x^{2}-2 x+1}{8 x^{2}+5}$$

Bridges to Calculus: An Introduction to Limits

Continuity and More on Limits

01:07

Precalculus

Evaluate the following limits. Write your answer in simplest form.
$$\lim _{h \rightarrow 0} \frac{\sqrt{x+h+2}-\sqrt{x+2}}{h}$$

Bridges to Calculus: An Introduction to Limits

Continuity and More on Limits

01:05

Precalculus

Evaluate the following limits. Write your answer in simplest form.
$$\lim _{h \rightarrow 0} \frac{\left[3(x+h)-(x+h)^{2}\right]-\left(3 x-x^{2}\right)}{h}$$

Bridges to Calculus: An Introduction to Limits

Continuity and More on Limits