Chapter 8

Use a CAS to evaluate the limits. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1-x-x^{2} / 2-x^{3} / 6}{x^{4}} $$

In electromagnetic theory, the magnetic potential \(u\) at a point on the axis of a circular coil is given by $$ u=A r \int_{a}^{\infty} \frac{d x}{\left(r^{2}+x^{2}\right)^{3 / 2}} $$ where \(A, r\), and \(a\) are constants. Evaluate \(u\).

Evaluate \(\int_{-3}^{3} \frac{x}{9-x^{2}} d x\) or show that it diverges.

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty}[\ln (x+1)-\ln (x-1)] $$

Use a CAS to evaluate the limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos \left(x^{2}\right)}{x^{3} \sin x} $$

There is a subtlety in the definition of \(\int_{-\infty}^{\infty} f(x) d x\) that is illustrated by the following: Show that (a) \(\int_{-\infty}^{\infty} \sin x d x\) diverges and (b) \(\lim _{a \rightarrow \infty} \int_{-a}^{a} \sin x d x=0\).

Evaluate \(\int_{-4}^{4} \frac{1}{16-x^{2}} d x\) or show that it diverges.

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} $$

Use a CAS to evaluate the limits. $$ \lim _{x \rightarrow 0} \frac{\tan x-x}{\arcsin x-x} $$

Consider an infinitely long wire coinciding with the positive \(x\) -axis and having mass density \(\delta(x)=\left(1+x^{2}\right)^{-1}\), \(0 \leq x<\infty\) (a) Calculate the total mass of the wire. (b) Show that this wire does not have a center of mass.

Evaluate \(\int_{-1}^{1} \frac{1}{x \sqrt{-\ln |x|}} d x\) or show that it diverges.

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}}(\ln x \cot x) $$

Plot the numerator \(f(x)\) and the denominator \(g(x)\) in the same graph window for each of these domains: \(-1 \leq x \leq 1,-0.1 \leq x \leq 0.1\), and \(-0.01 \leq x \leq 0.01\). From the plot, estimate the values of \(f^{\prime}(x)\) and \(g^{\prime}(x)\) and use these to approximate the given limit. $ \lim _{x \rightarrow 0} \frac{3 x-\sin x}{x} $$

Give an example of a region in the first quadrant that gives a solid of finite volume when revolved about the \(x\) -axis, but gives a solid of infinite volume when revolved about the \(y\) -axis.

If \(\lim _{x \rightarrow 0^{+}} f(x)=\infty\), we define $$ \int_{0}^{\infty} f(x) d x=\lim _{c \rightarrow 0^{-}} \int_{c}^{1} f(x) d x+\lim _{b \rightarrow \infty} \int_{1}^{b} f(x) d x $$ provided both limits exist. Otherwise, we say that \(\int_{0}^{\infty} f(x) d x\) diverges. Show that \(\int_{0}^{\infty} \frac{1}{x^{p}} d x\) diverges for all \(p\).

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{\int_{1}^{x} \sqrt{1+e^{-t}} d t}{x} $$

Let \(f\) be a nonnegative continuous function defined on \(0 \leq x<\infty\) with \(\int_{0}^{\infty} f(x) d x<\infty .\) Show that (a) if \(\lim _{x \rightarrow \infty} f(x)\) exists it must be 0 ; (b) it is possible that \(\lim _{x \rightarrow \infty} f(x)\) does not exist.

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 1^{+}} \frac{\int_{1}^{x} \sin t d t}{x-1} $$

We can use a computer to approximate \(\int_{1}^{\infty} f(x) d x\) by taking \(b\) very large in \(\int_{1}^{b} f(x) d x\) provided we know that the first integral converges. Calculate \(\int_{1}^{100}\left(1 / x^{p}\right) d x\) for \(p=2,1.1,1.01\), 1 , and \(0.99 .\) Note that this gives no hint that the integral \(\int_{1}^{\infty}\left(1 / x^{p}\right) d x\) converges for \(p>1\) and diverges for \(p \leq 1\)

Find the area of the region between the curves \(y=(x-8)^{-2 / 3}\) and \(y=0\) for \(0 \leq x<8\)

Find each limit. Hint: Transform to problems involving a continuous variable \(x\). Assume that \(a>0\). (a) \(\lim _{n \rightarrow \infty} \sqrt[n]{a}\) (b) \(\lim _{n \rightarrow \infty} \sqrt[n]{n}\) (c) \(\lim _{n \rightarrow \infty} n(\sqrt[n]{a}-1)\) (d) \(\lim _{n \rightarrow \infty} n(\sqrt[n]{n}-1)\)

Approximate \(\int_{0}^{a} \frac{1}{\pi}\left(1+x^{2}\right)^{-1} d x\) for \(a=10,50\), and \(100 .\)

Find the area of the region between the curves \(y=1 / x\) and \(y=1 /\left(x^{3}+x\right)\) for \(0

Find each limit. (a) \(\lim _{x \rightarrow 0^{+}} x^{x}\) (b) \(\lim _{x \rightarrow 0^{+}}\left(x^{x}\right)^{x}\) (c) \(\lim _{x \rightarrow 0^{+}} x^{\left(x^{x}\right)}\) (d) \(\lim _{x \rightarrow 0^{+}}\left(\left(x^{x}\right)^{x}\right)^{x}\) (e) \(\lim _{x \rightarrow 0^{+}} x^{\left(x^{\left(x^{r}\right)}\right)}\)

Approximate \(\int_{-a}^{a} \frac{1}{\sqrt{2 \pi}} \exp \left(-x^{2} / 2\right) d x\) for \(a=1,2,3\), and \(4 .\)

Let \(R\) be the region in the first quadrant below the curve \(y=x^{-2 / 3}\) and to the left of \(x=1\) (a) Show that the area of \(R\) is finite by finding its value. (b) Show that the volume of the solid generated by revolving \(R\) about the \(x\) -axis is infinite.

Graph \(y=x^{1 / x}\) for \(x>0 .\) Show what happens for very small \(x\) and very large \(x\). Indicate the maximum value.

Find \(b\) so that \(\int_{0}^{b} \ln x d x=0\).

Find each limit. (a) \(\lim _{x \rightarrow 0^{+}}\left(1^{x}+2^{x}\right)^{1 / x}\) (b) \(\lim _{x \rightarrow 0^{-}}\left(1^{x}+2^{x}\right)^{1 / x}\) (c) \(\lim _{x \rightarrow \infty}\left(1^{x}+2^{x}\right)^{1 / x}\) (d) \(\lim _{x \rightarrow-\infty}\left(1^{x}+2^{x}\right)^{1 / x}\)

(Comparison Test) If \(0 \leq f(x) \leq g(x)\) on \([a, \infty)\), it can be shown that the convergence of \(\int_{a}^{\infty} g(x) d x\) implies the convergence of \(\int_{a}^{\infty} f(x) d x\), and the divergence of \(\int_{a}^{\infty} f(x) d x\) implies the divergence of \(\int_{a}^{\infty} g(x) d x .\) Use this to show that \(\int_{1}^{\infty} \frac{1}{x^{4}\left(1+x^{4}\right)} d x\) converges. Hint: On \([1, \infty), 1 /\left[x^{4}\left(1+x^{4}\right)\right] \leq 1 / x^{4}\).

For \(k \geq 0\), find $$ \lim _{n \rightarrow \infty} \frac{1^{k}+2^{k}+\cdots+n^{k}}{n^{k+1}} $$ Hint: Though this has the \(\infty / \infty\) form, l'Hôpital's Rule is not helpful. Think of a Riemann sum.

Let \(c_{1}, c_{2}, \ldots, c_{n}\) be positive constants with \(\sum_{i=1}^{n} c_{i}=1\), and let \(x_{1}, x_{2}, \ldots, x_{n}\) be positive numbers. Take natural logarithms and then use l'Hôpital's Rule to show that $$ \lim _{t \rightarrow 0^{+}}\left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1 / t}=x_{1}^{c_{1}} x_{2}^{c_{2}} \cdots x_{n}^{c_{n}}=\prod_{i=1}^{n} x_{i}^{c_{i}} $$ Here \(\prod\) means product; that is, \(\prod_{i=1}^{n} a_{i}\) means \(a_{1} \cdot a_{2} \cdots \cdots a_{n} .\) In particular, if \(a, b, x\), and \(y\) are positive and \(a+b=1\), then $$ \lim _{t \rightarrow 0^{+}}\left(a x^{t}+b y^{t}\right)^{1 / t}=x^{a} y^{b} $$

Use the Comparison Test of Problem 46 to show that \(\int_{1}^{\infty} e^{-x^{2}} d x\) converges. Hint: \(e^{-x^{2}} \leq e^{-x}\) on \([1, \infty)\)

Consider \(f(x)=n^{2} x e^{-n x}\). (a) Graph \(f(x)\) for \(n=1,2,3,4,5,6\) on \([0,1]\) in the same graph window. (b) For \(x>0\), find \(\lim _{n \rightarrow \infty} f(x)\). (c) Evaluate \(\int_{0}^{1} f(x) d x\) for \(n=1,2,3,4,5,6\). (d) Guess at \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x) d x\). Then justify your answer rigorously.

Find the absolute maximum and minimum points (if they exist) for \(f(x)=\left(x^{25}+x^{3}+2^{x}\right) e^{-x}\) on \([0, \infty)\).

(Gamma Function) Let \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, n>0 .\) This integral converges by Problems 51 and \(52 .\) Show each of the following (note that the gamma function is defined for every positive real number \(n\) ): (a) \(\Gamma(1)=1\) (b) \(\Gamma(n+1)=n \Gamma(n)\) (c) \(\Gamma(n+1)=n !\), if \(n\) is a positive integer.

Evaluate \(\int_{0}^{\infty} x^{n-1} e^{-x} d x\) for \(n=1,2,3,4\), and 5, thereby confirming Problem \(53(\mathrm{c})\).

By interpreting each of the following integrals as an area and then calculating this area by a \(y\) -integration, evaluate: (a) \(\int_{0}^{1} \sqrt{\frac{1-x}{x}} d x\) (b) \(\int_{-1}^{1} \sqrt{\frac{1+x}{1-x}} d x\)

Suppose that \(0