Limits

Explain what it means for a function $f$ to be continuous at a point $x=c$, with a sentence that includes the words “approaches” and “value.”

In our proof that constant functions are continuous, we used the fact that given any $\in$ > 0, a choice of any δ > 0 will work in the formal definition of limit. Use a graph to explain why this makes intuitive sense. (This exercise depends on Section 1.3.)

In our proof that linear functions are continuous, we used the fact that given any $\in >0$ , the choice of δ = $\frac{\in }{\left|m\right|}$ will work in the formal definition of limit. Use a graph to explain why this makes intuitive sense. (This exercise depends on Section 1.3.)

Given the following function $f$ , define $f\left(1\right)$ so that $f$ is  continuous at x = 1, if possible:

$f\left(x\right)=\frac{x^2-2x+1}{x^2-6x+5}$

Given the following function $f$ , define $f\left(1\right)$ so that $f$ is continuous at $x=1$, if possible:

$f\left(x\right)=\left\{\begin{array}{ll}3x-1, if& x<1\\ x^2+1. if& x>1\end{array}\right.$

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f. 

$\underset{x\to -1^{-}}{\lim }f\left(x\right)=2, \underset{x\to -1^{+}}{\lim }f\left(x\right)=2, f(-1)=1.$

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f. 

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=2, \underset{x\to 2^{+}}{\lim }f\left(x\right)=1, f\left(2\right)=1.$

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f. 

$\underset{x\to 0^{-}}{\lim }f\left(x\right)=-1, \underset{x\to 0^{+}}{\lim }f\left(x\right)=1, f\left(0\right)=0.$

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.  

$\underset{x\to 2^{-}}{\lim }f\left(x\right)=-\infty , \underset{x\to 2^{+}}{\lim }f\left(x\right)=\infty , f\left(2\right)=3.$

State what it means for a function f to be continuous at a point x = c, in terms of the delta–epsilon definition of limit. 

State what it means for a function f to be left continuous at a point x = c, in terms of the delta–epsilon definition of limit. 

State what it means for a function f to be right continuous at a point x = c, in terms of the delta–epsilon definition of limit. 

Sketch a labeled graph of a function that satisfies the hypothesis of the Extreme Value Theorem, and illustrate on your graph that the conclusion of the Extreme Value Theorem follows. 

Sketch a labeled graph of a function that satisfies the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem follows.

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem does not necessarily hold. 

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Extreme Value Theorem, and illustrate on your graph that the conclusion of the Extreme Value Theorem does not necessarily hold.

Explain why the Intermediate Value Theorem allows us to say that a function can change sign only at discontinuities and zeroes. 

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities: 

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities:  

For each function f graphed in Exercises 23–26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements. 

For each function f graphed in Exercises 23–26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.  

For each function f graphed in Exercises 23–26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements. 

For each function f graphed in Exercises 23–26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.  

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

f is left continuous at x = 1 and right continuous at x = 1, but is not continuous at x = 1, and f(1) = −2. 

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not. 

f is left continuous at x = 2 but not continuous at x = 2, and f(2) = 3.

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

$f has an infinite discontinuity at 0 but is right continuous at 0, and f \left(0\right) = 1.$

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

$$\begin{gathered} f has a removable discontinuity at x = -2 and is right continuous \\ at x = -2, and f (-2) = 0. \end{gathered}$$

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

f is continuous on [0, 2) but not on [0, 2].

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to -1}{\lim } 6.$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to -1}{\lim } x.$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to -5}{\lim } (3x-2) .$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to 3}{\lim } x^4 .$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to 0}{\lim } x^{-3} .$

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to -5}{\lim } \sqrt{x} .$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right) = \left\{\begin{array}{l}(x-3) , if x<3\\ -(x-3) , if x⩾3\end{array}\right. .$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right) = \left\{\begin{array}{l}(x-3) , if x<0\\ -(x-3) , if x⩾0\end{array} .\right.$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right) = \left\{\begin{array}{c}{x}^2 , if x<2\\ 4 , if x=2\\ 2x+1 , if x>2\end{array} .\right.$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right) = \left\{\begin{array}{c}\sqrt{-x} , if x<0\\ 2 , if x=0\\ \sqrt{x} , if x>0\end{array}\right. .$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right) = \left\{\begin{array}{c}x+1, if x<1\\ 3x-1, if1⩽x<2 \\ x+2 , if x⩾2\end{array}\right. .$

Use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f , describe the type of discontinuity and any one-sided discontinuity. 

$f\left(x\right)=\left\{\begin{array}{r}{x}^3,\text{ if }x\le 0\\ 1-x,\text{ if }0<x<3\\ x-5,\text{ if }x\ge 3\end{array}\right.$

Use graphs to determine if each function is continuous at the given point x = c. 

$f\left(x\right)=\left\{\begin{array}{r}2-x,\text{ if }x\text{ rational }\\ x^2,\text{ if }x\text{ irrational, }\end{array}\right. c=2$

Use graphs to determine if each function is continuous at the given point x = c. 

$f\left(x\right)=\left\{\begin{array}{l}{x}^2-3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ 3x+1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational, }\end{array}c=0\right.$

Use graphs to determine if each function is continuous at the given point x = c.

 $f\left(x\right)=\left\{\begin{array}{r}2-x,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ x^2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational },\end{array}c=1\right.$

Use graphs to determine if each function is continuous at the given point x = c. 

$f\left(x\right)=\left\{\begin{array}{l}{x}^2-3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ 3x+1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational, }\end{array}c=4\right.$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere. 

$f\left(x\right)=x^4-3x^2-2, [a,b]=[-2,2]$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere. 

$f\left(x\right)=x^4-3x^2-2, [a,b]=[0,2]$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere. 

$f\left(x\right)=x^4-3x^2-2, [a,b]=[-1,1]$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere. 

$f\left(x\right)=3-2x^2+x^3, [a,b]=[-1,2]$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere. 

$f\left(x\right)=3-2x^2+x^3, [a,b]=[0,2]$