Chapter 13 Supplement Limits and Continuity to Accompany Second Edition Contemporary Precalculus

Problem 1

In Problems 1 through 8 the functions are continuous at the value $a$ given. In each case find a value $\delta$ corresponding to the given value of $\varepsilon$ so that the definition of continuity is satisfied. Draw a graph.
$$
f(x)=2 x+5, a=1, \varepsilon=0.01
$$

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

In Problems 9 through 17 the functions are defined in an interval about the given value of $a$ but not at $a$. Determine a value $\delta$ so that for the given values of $L$ and $\varepsilon$, the statement " $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$ " is valid. Sketch the graph of the given function.
$$
f(x)=\left(x^{2}-9\right) /(x+3), a=-3, L=-6, \varepsilon=0.005
$$

Problem 10

The functions are defined in an interval about the given value of $a$ but not at $a$. Determine a value $\delta$ so that for the given values of $L$ and $\varepsilon$, the statement " $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$ " is valid. Sketch the graph of the given function.
$$
f(x)=(\sqrt{2 x}-2) /(x-2), a=2, L=\frac{1}{2}, \varepsilon=0.01
$$

Problem 11

The functions are defined in an interval about the given value of $a$ but not at $a$. Determine a value $\delta$ so that for the given values of $L$ and $\varepsilon$, the statement " $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$ " is valid. Sketch the graph of the given function.
$$
f(x)=\left(x^{2}-4\right) /(x-2), a=2, L=4, \varepsilon=0.01
$$

Problem 12

Problem 13

The functions are defined in an interval about the given value of $a$ but not at $a$. Determine a value $\delta$ so that for the given values of $L$ and $\varepsilon$, the statement " $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$ " is valid. Sketch the graph of the given function.
$$
f(x)=\left(x^{3}-8\right) /(x-2), a=2, L=12, \varepsilon=0.5
$$

Problem 14

The functions are defined in an interval about the given value of $a$ but not at $a$. Determine a value $\delta$ so that for the given values of $L$ and $\varepsilon$, the statement " $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$ " is valid. Sketch the graph of the given function.
$$
f(x)=\left(x^{3}+1\right) /(x+1), a=-1, L=3, \varepsilon=0.1
$$

Problem 15

The functions are defined in an interval about the given value of $a$ but not at $a$. Determine a value $\delta$ so that for the given values of $L$ and $\varepsilon$, the statement " $|f(x)-L|<\varepsilon$ whenever $0<|x-a|<\delta$ " is valid. Sketch the graph of the given function.
$$
f(x)=(x-1) /(\sqrt[3]{x}-1), a=1, L=3, \varepsilon=0.1
$$

Problem 16

Problem 17

Problem 18

Show that $\lim _{x \rightarrow 0}(\sin (1 / x))$ does not exist.

Problem 19

Show that $\lim _{x \rightarrow 0} x \log |x|=0$.

Problem 20

The function $f(x)=x \cot x$ is not defined at $x=0$. Can the domain of $f$ be enlarged to include $x=0$ in such a way that the function is continuous on the enlarged domain?

Problem 21

For all $x \in R^{1}$ define
$$
f(x)= \begin{cases}1 & \text { if } x \text { is a rational number, } \\ 0 & \text { if } x \text { is an irrational number. }\end{cases}
$$
Show that $f$ is not continuous at every value of $x$.

Problem 22

Suppose that $f$ is defined in an interval about the number $a$ and $\lim _{h \rightarrow 0}[f(a+h)-$ $f(a-h)]=0$. Show that $f$ may not be continuous at $a$. Is it always true that $\lim _{h \rightarrow 0} f(a+h)$ exists?

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Source: https://www.numerade.com/books/chapter/continuity-and-limits-3/