Show that the limit is of indeterminate form and apply l'Hôpital's rule once.

Apply l'Hôpital's rule twice more to evaluate the limit.

Use the Maclaurin series for $e^x$ and $\sin x$ to confirm the limit.

Show that the limit is indeterminate and apply l'Hôpital’s rule twice.

Evaluate the limit by applying l'Hôpital’s rule a third time.

Derive the Maclaurin series for $\tan x$ up to $x^3$ and use it to confirm the limit.

Find ${\int }_1^R{x}^{-2}dx$, giving your answer in terms of $R$.

Hence show that ${\int }_1^{\infty }{x}^{-2}dx=1$.

Now consider ${\int }_1^{\infty }{x}^{-1/2}dx$.

Show that ${\int }_1^{\infty }{x}^{-1/2}dx$ does not converge to a finite value.

The results above suggest that the convergence of ${\int }_1^{\infty }{x}^{-p}dx$ depends on the value of $p$. In this section you will determine exactly when this integral converges.

For $p>0$ and $p\ne 1$, show that

$${\int }_1^R{x}^{-p}dx=\frac{R^{1-p}-1}{1-p}$$

Show that when $p>1$, ${\int }_1^{\infty }{x}^{-p}dx=\frac{1}{p-1}$.

Explain why ${\int }_1^{\infty }{x}^{-p}dx$ diverges when $0<p<1$.

Verify the result from part 5 by evaluating ${\int }_1^{\infty }{x}^{-3}dx$.

The function $f(x)=e^{-x^2}$ arises frequently in mathematics and statistics. It does not have an antiderivative that can be expressed in terms of elementary functions, so ${\int }_1^{\infty }{e}^{-x^2}dx$ cannot be evaluated using standard integration techniques. However, it is still possible to determine whether this integral converges.

Consider the inequality $e^{-x^2}\le e^{-x}$.

Show that $x^2\ge x$ for all $x\ge 1$.

Hence show that $e^{-x^2}\le e^{-x}$ for all $x\ge 1$.

Show that ${\int }_1^{\infty }{e}^{-x}dx=e^{-1}$.

Using the results from parts 9 and 10, explain why ${\int }_1^{\infty }{e}^{-x^2}dx$ must converge to a finite value.

The technique used in part 12, bounding an unknown integral by a known convergent one, can be applied more broadly. Consider the function

$$g(x)=\frac{1}{1+x^3}$$

By finding a suitable comparison function whose improper integral converges, show that ${\int }_1^{\infty }\frac{1}{1+x^3}dx$ converges. Determine an upper bound for the value of this integral.

Write down the composite function $(h\circ k)(x)$.

Determine the domain of $(h\circ k)(x)$. Exclude any $x$ for which $(h\circ k)(x)=\frac{1}{4}$.

Evaluate the limit ${\lim }_{x\to 1}(h\circ k)(x)$.

Now consider the function $f(x)=k(x)\sqrt{h(x)}$. Find $\int f(x)\mathrm{d}x$.

Show the limit is indeterminate and apply l'Hôpital's rule three times.

Evaluate the limit using l'Hôpital's rule a fourth time.

Use Maclaurin series for $\sin t$ and $e^t$ to confirm the limit.

Show that the function is increasing near $t=0$.