Chapter 2

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

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

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. $$\lim _{u \rightarrow-2} \frac{u^{2}-u x+2 u-2 x}{u^{2}-u-6}$$

Find the limits. \(\lim _{x \rightarrow \infty}\left(\sqrt{2 x^{2}+3}-\sqrt{2 x^{2}-5}\right)\)

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 0} \frac{(1-\cos x)^{2}}{x^{2}} $$

$$ \lim _{x \rightarrow 0} x^{4}=0 $$

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

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 1} \frac{t^{2}-1}{\sin (t-1)} $$

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. $$\lim _{x \rightarrow \pi} \frac{2 x^{2}-6 x \pi+4 \pi^{2}}{x^{2}-\pi^{2}}$$

Find the limits. \(\lim _{y \rightarrow-\infty} \frac{9 y^{3}+1}{y^{2}-2 y+2}\)

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow 3} \frac{x-\sin (x-3)-3}{x-3} $$

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 \(\lim _{x \rightarrow c} F(x)=0\).

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. $$\lim _{w \rightarrow-2} \frac{(w+2)\left(w^{2}-w-6\right)}{w^{2}+4 w+4}$$

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.

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow \pi} \frac{1+\sin (x-3 \pi / 2)}{x-\pi} $$

In Problems 24-35, at 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\()\). $$\lim _{x \rightarrow a} \sqrt{f^{2}(x)+g^{2}(x)}$$

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

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cot t}{1 / t} $$

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

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ h(\theta)=|\sin \theta+\cos \theta| $$

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

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

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

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow \pi / 4} \frac{(x-\pi / 4)^{2}}{(\tan x-1)^{2}} $$

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ r(\theta)=\tan \theta $$

Find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1\) (see Example 4\()\). $$\lim _{x \rightarrow a} \sqrt[3]{g(x)}[f(x)+3]$$

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

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{u \rightarrow \pi / 2} \frac{2-2 \sin u}{3 u} $$

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ f(u)=\frac{2 u+7}{\sqrt{u+5}} $$

Find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1\) (see Example 4\()\). $$\lim _{x \rightarrow a}[f(x)-3]^{4}$$

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

How long does it take money to double in value for the specified interest rate? (a) \(6 \%\) compounded monthly (b) \(6 \%\) compounded continuously

Suppose that \(\lim _{x \rightarrow a} f(x)=L\) and that \(f(a)\) exists (though it may be different from \(L\) ). Prove that \(f\) is bounded on some interval containing \(a\); that is, show that there is an interval \((c, d)\) with \(c

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ g(u)=\frac{u^{2}+|u-1|}{\sqrt[3]{u+1}} $$

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

Inflation between 1999 and 2004 ran at about \(2.5 \%\) per year. On this basis, what would you expect a car that would have cost \(\$ 20,000\) in 1999 to cost in 2004 ?

Prove that if \(f(x) \leq g(x)\) for all \(x\) in some deleted interval about \(a\) and if \(\lim _{x \rightarrow a} f(x)=L\) and \(\lim _{x \rightarrow a} g(x)=M\), then \(L \leq M\).

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ F(x)=\frac{1}{\sqrt{4+x^{2}}} $$

Find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1\) (see Example 4\()\). $$\lim _{u \rightarrow a}[f(u)+3 g(u)]^{3}$$

Find the limits. \(\lim _{x \rightarrow \sqrt[3]{5}} \frac{x^{2}}{5-x^{3}}\)

Manhattan Island is said to have been bought by Peter Minuit in 1626 for \(\$ 24\). Suppose that Minuit had instead put the \(\$ 24\) in the bank at \(6 \%\) interest compounded continuously. What would that \(\$ 24\) have been worth in 2000 ?

. Which of the following are equivalent to the definition of limit? (a) For some \(\varepsilon>0\) and every \(\delta>0,0<|x-c|<\delta \Rightarrow\) \(|f(x)-L|<\varepsilon .\) (b) For every \(\delta>0\), there is a corresponding \(\varepsilon>0\) such that \(0<|x-c|<\varepsilon \Rightarrow|f(x)-L|<\delta\) (c) For every positive integer \(N\), there is a corresponding positive integer \(M\) such that \(0<|x-c|<1 / M \Rightarrow|f(x)-L|\) \(<1 / N\). (d) For every \(\varepsilon>0\), there is a corresponding \(\delta>0\) such that \(0<|x-c|<\delta\) and \(|f(x)-L|<\varepsilon\) for some \(x\).

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ G(x)=\frac{1}{\sqrt{4-x^{2}}} $$

Find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(f\). $$f(x)=3 x^{2}$$

Find the limits. \(\lim _{x \rightarrow 5} \frac{x^{2}}{(x-5)(3-x)}\)

If Methuselah's parents had put \(\$ 100\) in the bank for him at birth and he left it there, what would Methuselah have had at his death ( 969 years later) if interest was \(4 \%\) compounded annually?

$$ \text { State in } \varepsilon-\delta \text { language what it means to say } \lim f(x) \neq L \text {. } $$

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ f(x)= \begin{cases}x & \text { if } x<0 \\ x^{2} & \text { if } 0 \leq x \leq 1 \\ 2-x & \text { if } x>1\end{cases} $$