Chapter 13

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4}(f(x)+g(x))$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow-1} \frac{x^{6}-1}{x^{4}-1}$$

Use a calculator to estimate the limit. $$\lim _{x \rightarrow-\infty}[\sqrt{x^{2}+x+1}+x]$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow 2} \frac{x^{5}-32}{x^{3}-8}$$

Use a calculator to estimate the limit. $$\lim _{x \rightarrow \infty} \frac{x^{5 / 4}+x}{2 x-x^{5 / 4}}$$

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} \frac{g(x)}{h(x)}$$

Use a calculator to estimate the limit. $$\lim _{x \rightarrow-\infty} \sin \frac{1}{x}$$

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} f(x) g(x)$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow 0} \frac{x+\sin 2 x}{x-\sin 2 x}$$

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} h(x)^{2}$$

Use a calculator to estimate the limit. $$\lim _{x \rightarrow \infty} \frac{\ln x}{x}$$

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} \frac{3 h(x)}{2 f(x)+g(x)}$$

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$f(x)=x^{2}+5(x-2)^{7} \quad \text { at } x=3$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{x}$$

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} \frac{f(x)-2 g(x)}{4 h(x)}$$

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$g(x)=\left(x^{2}-3 x-10\right)\left(x^{3}+2 x^{2}-5 x+4\right) \quad \text { at } x=-1$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2}\left(6 x^{3}-2 x^{2}+5 x-3\right)$$

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$f(x)=\frac{x^{2}-9}{\left(x^{2}-x-6\right)\left(x^{2}+6 x+9\right)} \quad \text { at } x=2$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow 0}\left(x \sin \frac{1}{x}\right)$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-1}\left(x^{7}+2 x^{5}-x^{4}+3 x+4\right)$$

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$h(x)=\frac{x+3}{\left(x^{2}-x-1\right)\left(x^{2}+1\right)} \quad \text { at } x=-2$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow \pi^{-}} \frac{\sin x}{1-\cos x}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-2} \frac{3 x-1}{2 x+3}$$

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$f(x)=\frac{x \sqrt{x}}{(x-6)^{2}} \quad \text { at } x=36$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow \frac{\pi}{2}^{-}}(\sec x-\tan x)$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3} \frac{x^{2}+x+1}{x^{2}-2 x}$$

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. $$k(x)=\frac{\sqrt{8-x^{2}}}{2 x^{2}-5} \quad \text { at } x=-2$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x^{2}-2 x-3}$$

Explain why the function is not continuous at the given number. $$f(x)=1 /(x-3)^{3} \quad \text { at } x=3$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{2}+x-2}$$

Explain why the function is not continuous at the given number. $$h(x)=\frac{x^{2}+4}{x^{2}-x-2} \quad \text { at } x=2$$

Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$f(x)=\frac{3 x^{2}+5}{4 x^{2}-6 x+2}$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow 0^{-}} \frac{\sin (6 x)}{x}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1}$$

Explain why the function is not continuous at the given number. $$f(x)=\frac{x^{2}+4 x+3}{x^{2}-x-2} \quad \text { at } x=-1$$

Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$g(x)=\frac{x^{2}}{x^{2}-2 x+1}$$

Use the table feature of your calculator to find the limit. $$\lim _{x \rightarrow 0^{+}} \frac{\sin (3 x)}{1+\sin (4 x)}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-2} \frac{x^{2}+5 x+6}{x^{2}-x-6}$$

Explain why the function is not continuous at the given number. $$g(x)=\left\\{\begin{array}{ll} \sin (\pi / x) & \text { if } x \neq 0 \\ 1 & \text { if } x=0 \end{array} \text { at } x=0\right.$$

Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$h(x)=\frac{2 x^{2}-6 x+1}{2+x-x^{2}}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 4^{-}} \frac{x-4}{x^{2}-16}$$

Explain why the function is not continuous at the given number. $$f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \neq 0 \\ 1 & \text { if } x=0 \end{array} \text { at } x=0\right.$$

Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$k(x)=\frac{3 x+x^{2}-4}{2 x-x^{3}+x^{2}}$$

Explain why the function is not continuous at the given number. $$f(x)=\frac{\sqrt{2+x}-\sqrt{2}}{x} \text { at } x=0$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2^{+}} \frac{|x-2|}{x-2}$$

Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$f(x)=\frac{3 x^{4}-2 x^{3}+5 x^{2}-x+1}{7 x^{3}-4 x^{2}+6 x-12}$$

Determine whether or not the function is continuous at the given number. $$f(x)=\left\\{\begin{array}{cl} -2 x+4 & \text { if } x \leq 2 \\ 2 x-4 & \text { if } x>2 \end{array} \text { at } x=2\right.$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3^{+}} \frac{3}{x^{2}-9}$$

Use the Infinite Limit Theorem and the properties of limits as in Example 6 to find the horizontal asymptotes (if any) of the graph of the given function. $$g(x)=\frac{2 x^{5}-x^{3}+2 x-9}{5-x^{5}}$$

Determine whether or not the function is continuous at the given number. $$g(x)=\left\\{\begin{array}{cl} 2 x+5 & \text { if } x<-1 \\ -2 x+1 & \text { if } x \geq-1 \end{array} \text { at } x=-1\right.$$