5.3.E: Problems on \(L^{\prime}\) Hôpital's Rule

Elementary differentiation formulas are assumed known.

Exercise \(\PageIndex{1}\)

Complete the proof of L'Hôpital's rule. Verify that the differentiability assumption may be replaced by continuity plus existence of finite or infinite (but not both together infinite) derivatives \(f^{\prime}\) and \(g^{\prime}\) on \(G_{\neg p}\) (same proof).

Exercise \(\PageIndex{2}\)

Show that the rule fails for complex functions. See, however, Problems \(3,\) \(7,\) and \(8 .\)
[Hint: Take \(p=0\) with
\[
f(x)=x \text { and } g(x)=x+x^{2} e^{i / x^{2}}=x+x^{2}\left(\cos \frac{1}{x^{2}}+i \cdot \sin \frac{1}{x^{2}}\right) .
\]
Then
\[
\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}=1, \text { though } \lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lim _{x \rightarrow 0} \frac{1}{g^{\prime}(x)}=0 .
\]
Indeed, \(g^{\prime}(x)-1=(2 x-2 i / x) e^{i / x^{2}} .\) (Verify!) Hence
\[
\left.\left|g^{\prime}(x)\right|+1 \geq|2 x-2 i / x| \quad \text { (for }\left|e^{i / x^{2}}\right|=1\right) ,
\]
so
\[
\left|g^{\prime}(x)\right| \geq-1+\frac{2}{x} . \quad(\text { Why? })
\]
Deduce that
\[
\left.\left|\frac{1}{g^{\prime}(x)}\right| \leq\left|\frac{x}{2-x}\right| \rightarrow 0 .\right]
\]

Exercise \(\PageIndex{3}\)

Prove the "simplified rule of \(L^{\prime}\) Hôpital" for real or complex functions \(\text { (also for vector-valued } f \text { and scalar-valued } g) :\) If \(f\) and \(g\) are differentiable at \(p,\) with \(g^{\prime}(p) \neq 0\) and \(f(p)=g(p)=0\), then
\[
\lim _{x \rightarrow p} \frac{f(x)}{g(x)}=\frac{f^{\prime}(p)}{g^{\prime}(p)} .
\]
[Hint:
\[
\frac{f(x)}{g(x)}=\frac{f(x)-f(p)}{g(x)-g(p)}=\frac{\Delta f}{\Delta x} / \frac{\Delta g}{\Delta x} \rightarrow \frac{f^{\prime}(p)}{g^{\prime}(p)} .
\]

Exercise \(\PageIndex{4}\)

Why does \(\lim _{x \rightarrow+\infty} \frac{f(x)}{g(x)}\) not exist, though \(\lim _{x \rightarrow+\infty} \frac{f^{\prime}(x)}{g^{\prime}(x)}\) does, in the following example? Verify and explain.
\[
f(x)=e^{-2 x}(\cos x+2 \sin x), \quad g(x)=e^{-x}(\cos x+\sin x) .
\]
[Hint: \(g^{\prime}\) vanishes many times in each \(G_{+\infty} .\) Use the Darboux property for the \(\text { proof. }]\)

Exercise \(\PageIndex{5}\)

Find \(\lim _{x \rightarrow 0^{+}} \frac{e^{-1 / x}}{x}\).
\(\left.\text { [Hint: Substitute } z=\frac{1}{x} \rightarrow+\infty . \text { Then use the rule. }\right]\)

Exercise \(\PageIndex{6}\)

Verify that the assumptions of L'Hôpital's rule hold, and find the following limits.
(a) \(\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{\ln (e-x)+x-1}\);
(b) \(\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}-2 x}{x-\sin x}\);
(c) \(\lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e}{x}\);
(d) \(\lim _{x \rightarrow 0^{+}}\left(x^{q} \ln x\right), q>0\);
(e) \(\lim _{x \rightarrow+\infty}\left(x^{-q} \ln x\right), q>0\);
(f) \(\lim _{x \rightarrow 0^{+}} x^{x}\);
(g) \(\lim _{x \rightarrow+\infty}\left(x^{q} a^{-x}\right), a>1, q>0\);
(h) \(\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\operatorname{cotan}^{2} x\right)\);
(i) \(\lim _{x \rightarrow+\infty}\left(\frac{\pi}{2}-\arctan x\right)^{1 / \ln x}\);
(j) \(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{1 /(1-\cos x)}\).

Exercise \(\PageIndex{7}\)

Prove L'Hôpital's rule for \(f : E^{1} \rightarrow E^{n}(C)\) and \(g : E^{1} \rightarrow E^{1},\) with
\[
\lim _{k \rightarrow p}|f(x)|=0=\lim _{x \rightarrow p}|g(x)|, p \in E^{*} \text { and } r \in E^{n} ,
\]
leaving the other assumptions unchanged.
\(\left.\left.\text { [Hint: Apply the rule to the components of } \frac{f}{g} \text { (respectively, to }\left(\frac{f}{g}\right)_{\mathrm{re}} \text { and }\left(\frac{f}{g}\right)_{\mathrm{im}}\right) .\right]\)

Exercise \(\PageIndex{8}\)

Let \(f\) and \(g\) be complex and differentiable on \(G_{\neg p}, p \in E^{1} .\) Let
\[
\lim _{x \rightarrow p} f(x)=\lim _{x \rightarrow p} g(x)=0, \lim _{x \rightarrow p} f^{\prime}(x)=q, \text { and } \lim _{x \rightarrow p} g^{\prime}(x)=r \neq 0 .
\]
Prove that \(\lim _{x \rightarrow p} \frac{f(x)}{g(x)}=\frac{q}{r}\).
[Hint:
\[
\frac{f(x)}{g(x)}=\frac{f(x)}{x-p} / \frac{g(x)}{x-p} .
\]
Apply Problem 7 to find
\[
\lim _{x \rightarrow p} \frac{f(x)}{x-p} \text { and } \lim _{x \rightarrow p} \frac{g(x)}{x-p} .]
\]

Exercise \(\PageIndex{*9}\)

Do Problem 8 for \(f : E^{1} \rightarrow C^{n}\) and \(g : E^{1} \rightarrow C\).