Chapter 2

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

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

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 in- terval containing \(a\); that is, show that there is an interval \((c, d)\) with \(c

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

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 ?

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

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 \(L \leq M\)

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

What points, if any, are the functions discontinuous? $$ G(x)=\frac{1}{\sqrt{4-x^{2}}} $$

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 ?

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

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 \(x\)

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

What points, if any, are the functions discontinuous? $$ f(x)=\left\\{\begin{array}{ll} x & \text { if } x<0 \\ x^{2} & \text { if } 0 \leq x \leq 1 \\ 2-x & \text { if } x>1 \end{array}\right. $$

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?

Find the limits. $$ \lim _{\theta \rightarrow \pi^{+}} \frac{\theta^{2}}{\sin \theta} $$

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

, find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(\bar{f}\) $$ f(x)=3 x^{2}+2 x+1 $$

What points, if any, are the functions discontinuous? $$ g(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x<0 \\ -x & \text { if } 0 \leq x \leq 1 \\ x & \text { if } x>1 \end{array}\right. $$

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{5} 12 $$

Find the limits. $$ \lim _{x \rightarrow 3^{-}} \frac{x^{3}}{x-3} $$

Sketch the graph of $$ f(x)=\left\\{\begin{aligned} -x & \text { if } x<0 \\ x & \text { if } 0 \leq x<1 \\ 1+x & \text { if } x \geq 1 \end{aligned}\right. $$ Then find each of the following or state that it does not exist. (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 1} f(x)\) (c) \(f(1)\) (d) \(\lim _{x \rightarrow 1^{+}} f(x)\)

, find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(\bar{f}\) $$ f(x)=\frac{1}{x} $$

What points, if any, are the functions discontinuous? $$ f(t)=[t] $$

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{7}(0.11) $$

Find the limits. $$ \lim _{\theta \rightarrow(\pi / 2)^{+}} \frac{\pi \theta}{\cos \theta} $$

What points, if any, are the functions discontinuous? $$ g(t)=\left[t+\frac{1}{2}\right] $$

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{11}(8.12)^{1 / 5} $$

Find the limits. $$ \lim _{x \rightarrow 3^{-}} \frac{x^{2}-x-6}{x-3} $$

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{10}(8.57)^{7} $$

Find the limits. $$ \lim _{x \rightarrow 2^{+}} \frac{x^{2}+2 x-8}{x^{2}-4} $$

Sketch the graph of a function that has domain \([0,2]\) and is continuous on \([0,2)\) but not on \([0,2]\).

Use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take \(\ln\) of both sides, obtaining \(x \ln 3=\ln 11 ;\) then \(x=(\ln 11) /(\ln 3) \approx 2.1827 .\) $$ 2^{x}=17 $$

$$ \text { Find } \lim _{x \rightarrow 1}\left(x^{2}-1\right) /|x-1| \text { or state that it does not exist. } $$

Prove that \(\lim _{x \rightarrow c} f(x)=L \Leftrightarrow \lim _{x \rightarrow c}[f(x)-L]=0\).

Sketch the graph of a function that has domain \([0,6]\) and is continuous on \([0,2]\) and \((2,6]\) but is not continuous on \([0,6]\).

Use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take \(\ln\) of both sides, obtaining \(x \ln 3=\ln 11 ;\) then \(x=(\ln 11) /(\ln 3) \approx 2.1827 .\) $$ 5^{x}=13 $$

. Evaluate \(\lim _{x \rightarrow 0}(\sqrt{x+2}-\sqrt{2}) / x .\) Hint: Rationalize the numerator by multiplying the numerator and denominator by \(\sqrt{x+2}+\sqrt{2}\)

Prove that \(\lim _{x \rightarrow c} f(x)=0 \Leftrightarrow \lim _{x \rightarrow c}|f(x)|=0\).

Sketch the graph of a function that has domain \([0,6]\) and is continuous on \((0,6)\) but not on \([0,6]\).

Use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take \(\ln\) of both sides, obtaining \(x \ln 3=\ln 11 ;\) then \(x=(\ln 11) /(\ln 3) \approx 2.1827 .\) $$ 5^{2 s-3}=4 $$

Find the limits. $$ \lim _{x \rightarrow 0^{-}} \frac{|x|}{x} $$

. Let $$ f(x)=\left\\{\begin{aligned} x & \text { if } x \text { is rational } \\ -x & \text { if } x \text { is irrational } \end{aligned}\right. $$ Find each value, if possible. (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 0} f(x)\)

$$ \text { . Prove that } \lim _{x \rightarrow c}|x|=|c| \text { . } $$

Let $$ f(x)=\left\\{\begin{aligned} x & \text { if } x \text { is rational } \\ -x & \text { if } x \text { is irrational } \end{aligned}\right. $$ Sketch the graph of this function as best you can and decide where it is continuous.

Use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take \(\ln\) of both sides, obtaining \(x \ln 3=\ln 11 ;\) then \(x=(\ln 11) /(\ln 3) \approx 2.1827 .\) $$ 12^{1 /(\theta-1)}=4 $$

Find the limits. $$ \lim _{x \rightarrow 0^{+}} \frac{|x|}{x} $$

Sketch, as best you can, the graph of a function \(f\) that satisfies all the following conditions. (a) Its domain is the interval \([0,4]\). (b) \(f(0)=f(1)=f(2)=f(3)=f(4)=1\) (c) \(\lim _{x \rightarrow 1} f(x)=2\) (d) \(\lim _{x \rightarrow 2} f(x)=1\) (e) \(\lim _{r \rightarrow 3^{-}} f(x)=2\) (f) \(\lim _{\rightarrow+} f(x)=1\)

In Problems \(41-48\), determine whether the function is continuous at the given point \(c .\) If the function is not continuous, determine whether the discontinuity is removable or non removable. $$ f(x)=\sin x ; c=0 $$

Verify that the given equations are identities. \(e^{x}=\cosh x+\sinh x\)