Chapter 2

Determine the one-sided limit. $$ \lim _{x \rightarrow-2^{+}}\left(x^{3}+3 x-5\right) $$

Determine whether \(f\) is continuous at \(a\). $$ f(x)=\sqrt{x} \cos x ; a=\frac{\pi}{4} $$

In Exercises \(1-8\) guess the value of the limit. \(\lim _{x \rightarrow-1}(x+4)\)

Use the results of this section to evaluate the given limit. $$ \lim _{x \rightarrow 2}(-5) $$

Determine the one-sided limit. $$ \lim _{x \rightarrow \pi / 3^{-}} \cos (x+\pi / 6) $$

Determine whether \(f\) is continuous at \(a\). $$ f(x)=e^{x} \ln x ; a=1 $$

Guess the value of the limit. \(\lim _{x \rightarrow 5}(-2 x+7)\)

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow \sqrt{2}}\left(x^{2}+5\right)(\sqrt{2} x+1) $$

Determine the one-sided limit. $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}-4}{x-2} $$

Determine whether \(f\) is continuous at \(a\). $$ f(x)=\left\\{\begin{array}{ll} \frac{\sin x}{x} & \text { for } x \neq 0 \\ 1 & \text { for } x=0 \end{array} \quad a=0\right. $$

Guess the value of the limit. \(\lim _{h \rightarrow 0}\left(2 h+h^{2}\right)\)

Use the results of this section to evaluate the limit. $$ \lim _{y \rightarrow 64}(\sqrt[3]{y}+\sqrt{y})^{2} $$

Determine the one-sided limit. $$ \lim _{x \rightarrow 2^{+}} \frac{x^{2}-4}{x-2} $$

Determine whether \(f\) is continuous at \(a\). $$ f(x)=\left\\{\begin{array}{ll} \frac{\sin 5 x}{x} & \text { for } x \neq 0 \\ 1 & \text { for } x=0 \end{array} \quad a=0\right. $$

Guess the value of the limit. \(\lim _{h \rightarrow 0}\left(1-\frac{h^{2}}{2}\right)\)

Use the results of this section to evaluate the limit. $$ \lim _{t \rightarrow 0} \frac{2 t^{1 / 3}-4}{-3 t^{1 / 3}+5} $$

Determine the one-sided limit. $$ \lim _{x \rightarrow 1^{+}} \frac{x^{2}+3 x-4}{x^{2}-1} $$

Guess the value of the limit. \(\lim _{x \rightarrow 2} \frac{2 x-5}{4 x+3}\)

Use the results of this section to evaluate the given limit. $$ \lim _{x \rightarrow-1}(2 x+5) $$

Determine the one-sided limit. $$ \lim _{x \rightarrow-2^{-}} \frac{x^{2}-3 x-10}{x^{2}-9} $$

For which of the following functions can we define \(f(a)\) so as to make \(f\) continuous at \(a\) ? For those we can, find the value of \(f(a)\) that makes \(f\) continuous at \(a\). a. \(f(x)=\frac{x^{2}-9}{x-3} ; a=3\) b. \(f(x)=\frac{x^{2}+5 x+4}{x-1} ; a=1\) c. \(f(x)=\frac{e^{x}-1}{x} ; a=0\) d. \(f(x)=\sin \frac{1}{x} ; a=0\)

Guess the value of the limit. \(\lim _{x \rightarrow 1 / 2} \frac{3 x-2}{4 x-1}\)

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow-\pi / 3} 3 x^{2} \cos x $$

Use the results of this section to evaluate the given limit. $$ \lim _{x \rightarrow 0}\left(-4 x-\frac{1}{3}\right) $$

Determine the one-sided limit. $$ \lim _{t \rightarrow 5^{+}} \frac{|t-5|}{5-t} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\sqrt{x-2} ; a=2 $$

Guess the value of the limit. \(\lim _{x \rightarrow 3} \frac{\sqrt{x+1}}{2 x-1}\)

Determine the one-sided limit. $$ \lim _{t \rightarrow-3^{+}} \sqrt{t+3} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\ln \left(x^{2}+1\right) ; a=0 $$

Guess the value of the limit. \(\lim _{x \rightarrow-1 / 2} \frac{1}{2} \sqrt{\frac{1}{x}+6}\)

Determine the infinite limit. $$ \lim _{x \rightarrow 0^{-}} 4 / x^{3} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\sqrt{1-e^{x}} ; a=0 $$

Use the results of this section to evaluate the limit. $$ \lim _{y \rightarrow 2 \pi / 3} \frac{\pi \sin y \cos y}{y} $$

Determine the infinite limit. $$ \lim _{x \rightarrow 0^{+}} 4 / x^{3} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\tan \sqrt{x-\pi / 2} ; a=\pi / 2 $$

First simplify the given expression and then guess the value of the limit. \(\lim _{x \rightarrow 1} \frac{x^{2}+4 x-5}{x-1}\)

Use the results of this section to evaluate the given limit. $$ \lim _{x \rightarrow 0} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} -\frac{3}{2} x+\frac{1}{4} & \text { for } x \neq \frac{1}{2} \\ 0 & \text { for } x=0 \end{array}\right. $$

Determine the infinite limit. $$ \lim _{x \rightarrow 0}-1 / x^{2} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\left\\{\begin{array}{ll} \frac{|x-4|}{x-4} & \text { for } x \neq 4 \\ e^{4-x} & \text { for } x=4 \end{array} \quad a=0,4\right. $$

First simplify the given expression and then guess the value of the limit. \(\lim _{x \rightarrow-5} \frac{x^{2}+4 x-5}{x+5}\)

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow \sqrt{5}}\left(9-x^{2}\right)^{-5 / 2} $$

Use the definition of limit to verify the given limit. $$ \lim _{x \rightarrow 0}(4 x+7)=7 $$

Determine the infinite limit. $$ \lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 4}} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(x)=\left\\{\begin{array}{rl} -1 & \text { for } x<0 \\ 0 & \text { for } x=0 \\ 1 & \text { for } x>0 \end{array} \quad a=0\right. $$

First simplify the given expression and then guess the value of the limit. \(\lim _{x \rightarrow \pi} \frac{x^{2}-2 x+1}{(x-1)^{2}}\)

Use the results of this section to evaluate the limit. $$ \lim _{t \rightarrow 3 \pi / 2} \sin \left(\frac{\pi}{2} \sin t\right) $$

Use the definition of limit to verify the given limit. $$ \lim _{x \rightarrow 3}(-2 x+5)=-1 $$

Determine the infinite limit. $$ \lim _{y \rightarrow-1^{-}} \frac{\pi}{y+1} $$

Determine whether \(f\) is continuous or discontinuous at \(a\). If \(f\) is discontinuous at \(a\), determine whether \(f\) is continuous from the right at \(a\), continuous from the left at \(a\), or neither. $$ f(t)=t^{2} \sqrt{t^{2}-t^{4}} ; a=0,1 $$

First simplify the given expression and then guess the value of the limit. \(\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}\)