Chapter 2

Find the limits. $$ \lim _{n \rightarrow \infty} \frac{n}{n^{2}+1} $$

Use the fact that \(e=\lim _{h \rightarrow 0}(1+h)^{1 / h}\) to find each limit. (a) \(\lim _{x \rightarrow 0}(1-x)^{1 / x}\) Hint: \((1-x)^{1 / x}=\left[(1-x)^{1 /(-x)}\right]^{-1}\) (b) \(\lim _{x \rightarrow 0}(1+3 x)^{1 / x}\) (c) \(\lim _{n \rightarrow \infty}\left(\frac{n+2}{n}\right)^{n}\) (d) \(\lim _{n \rightarrow \infty}\left(\frac{n-1}{n}\right)^{2 n}\)

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x^{2}-1} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 1} \frac{10 x^{3}-26 x^{2}+22 x-6}{(x-1)^{2}}=4 $$

GC In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{\sin x}{2 x} $$

Plot the functions \(u(x), l(x)\), and \(f(x) .\) Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=2, l(x)=2-x^{2}, f(x)=1+\frac{\sin x}{x} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{2 x+1}{\sqrt{x^{2}+3}}\). Hint: Divide numerator and denominator \(x\). Note that, for \(x>0, \sqrt{x^{2}+3} / x=\sqrt{\left(x^{2}+3\right) / x^{2}}\). py

Find each of the following limits. (a) \(\lim _{n \rightarrow \infty}\left(1+\frac{2}{n}\right)^{100}\) (b) \(\lim _{n \rightarrow \infty}(1.001)^{n}\) (c) \(\lim _{n \rightarrow \infty}\left(\frac{n+3}{n}\right)^{n+1}\) (d) \(\lim _{x \rightarrow 0}(1+x)^{1 / x}\)

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{x \rightarrow-3} \frac{x^{2}-14 x-51}{x^{2}-4 x-21} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 1}\left(2 x^{2}+1\right)=3 $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cos t}{2 t} $$

Find the limits. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{2 x+1}}{x+4} $$

The given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? $$ H(t)=\frac{\sqrt{t}-1}{t-1} $$

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{u \rightarrow-2} \frac{u^{2}-u x+2 u-2 x}{u^{2}-u-6} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow-1}\left(x^{2}-2 x-1\right)=2 $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{(x-\sin x)^{2}}{x^{2}} $$

Find the limits. $$ \begin{array}{l} \underline{\phantom{xxx}} \lim _{x \rightarrow \infty}\left(\sqrt{2 x^{2}+3}-\sqrt{2 x^{2}-5}\right) . \text { Hint: } \quad \text { Multiply and }\\\ \text { divide by } \sqrt{2 x^{2}+3}+\sqrt{2 x^{2}-5} \end{array} $$

The given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? $$ \phi(x)=\frac{x^{4}+2 x^{2}-3}{x+1} $$

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 0} x^{4}=0 $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{(1-\cos x)^{2}}{x^{2}} $$

Find the limits. $$ \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+2 x}-x\right) $$

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{x \rightarrow \pi} \frac{2 x^{2}-6 x \pi+4 \pi^{2}}{x^{2}-\pi^{2}} $$

$$ \begin{array}{l} \text { 23. Prove that if } \lim _{x \rightarrow c} f(x)=L \text { and } \lim _{x \rightarrow c} f(x)=M, \text { then }\\\ L=M \end{array} $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 1} \frac{t^{2}-1}{\sin (t-1)} $$

Find the limits. \(\lim _{y \rightarrow-\infty} \frac{9 y^{3}+1}{y^{2}-2 y+2} .\) Hint: Divide numerator and denominator by \(y^{2}\).

In Problems \(24-35\), at what points, if any, are the functions discontinuous? $$ f(x)=\frac{3 x+7}{(x-30)(x-\pi)} $$

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{w \rightarrow-2} \frac{(w+2)\left(w^{2}-w-6\right)}{w^{2}+4 w+4} $$

Let \(F\) and \(G\) be functions such that \(0 \leq F(x) \leq G(x)\) for all \(x\) near \(c\), except possibly at \(c\). Prove that if \(\lim _{x \rightarrow c} G(x)=0\), then \(\lim _{x \rightarrow c} F(x)=0 .\)

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 3} \frac{x-\sin (x-3)-3}{x-3} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{a_{0} x^{n}+a_{1} x^{n-1}+\cdots+a_{n-1} x+a_{n}}{b_{0} x^{n}+b_{1} x^{n-1}+\cdots+b_{n-1} x+b_{n}}\), where \(a_{0} \neq 0\), \(b_{0} \neq 0\), and \(n\) is a natural number.

What points, if any, are the functions discontinuous? $$ f(x)=\frac{33-x^{2}}{x \pi+3 x-3 \pi-x^{2}} $$

, find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1(\) see Example 4) \(.\) $$ \lim _{x \rightarrow a} \sqrt{f^{2}(x)+g^{2}(x)} $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow \pi} \frac{1+\sin (x-3 \pi / 2)}{x-\pi} $$

Find the limits. $$ \lim _{n \rightarrow \infty} \frac{n}{\sqrt{n^{2}+1}} $$

What points, if any, are the functions discontinuous? $$ h(\theta)=|\sin \theta+\cos \theta| $$

Find the limits. $$ \lim _{n \rightarrow \infty} \frac{n^{2}}{\sqrt{n^{3}+2 n+1}} $$

, find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1(\) see Example 4) \(.\) $$ \lim _{x \rightarrow a} \frac{2 f(x)-3 g(x)}{f(x)+g(x)} $$

$$ \text { Prove that } \lim _{x \rightarrow 0^{+}} \sqrt{x}=0 \text { . } $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cot t}{1 / t} $$

What points, if any, are the functions discontinuous? $$ r(\theta)=\tan \theta $$

If \(\$ 375\) is put in the bank today, what will it be worth at the end of 2 years if interest is \(3.5 \%\) and is compounded as specified? (a) Annually (b) Monthly (c) Daily (d) Continuously

Find the limits. $$ \lim _{x \rightarrow 4^{+}} \frac{x}{x-4} $$

$$ \begin{array}{l} \text { By considering left- and right-hand limits, prove that }\\\ \lim _{x}|x|=0 \end{array} $$

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow \pi / 4} \frac{(x-\pi / 4)^{2}}{(\tan x-1)^{2}} $$

What points, if any, are the functions discontinuous? $$ f(u)=\frac{2 u+7}{\sqrt{u+5}} $$

Find the limits. $$ \lim _{t \rightarrow-3^{+}} \frac{t^{2}-9}{t+3} $$

$$ \begin{array}{l} \text { Prove that if }|f(x)|