Exercises

01

L’Hopital practice

Compute these limits using L’Hopital’s Rule:

• (a) ${\lim }_{x\to 4}\frac{x^2-2x-8}{x-4}$
• (b) ${\lim }_{x\to 0}\frac{e^x-1-x}{x^2}$
• (c) ${\lim }_{t\to \infty }\frac{\ln \left(\ln t\right)}{\ln t}$
• (d) ${\lim }_{x\to \infty }\frac{e^x}{x^n}$

02

L’Hopital practice

Compute these limits using L’Hopital’s Rule:

• (a) ${\lim }_{x\to (\pi /2{)}^{+}}\frac{\cos x}{1-\sin x}$
• (b) ${\lim }_{x\to \infty }x\sin \left(\frac{6\pi }{x}\right)$
• (c) ${\lim }_{x\to 1}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$
• (d) ${\lim }_{x\to \infty }\ln \left({\left(\frac{x}{5}\right)}^{2/x}\cdot \frac{4}{5}\right)$

03

L’Hopital mistake

What is wrong with the following “application” of L’Hopital’s Rule?
$$\begin{array}{c}\underset{x\to 1}{\lim }\frac{x^2-1}{x-1}\gg \gg \underset{x\to 1}{\lim }\frac{2x\cdot (x-1)-1\cdot (x^2-1)}{(x-1{)}^2}\\ \\ \gg \gg \underset{x\to 1}{\lim }\frac{2x^2-2x-x^2+1}{(x-1{)}^2}\gg \gg \underset{x\to 1}{\lim }\frac{x^2-2x+1}{(x-1{)}^2}\\ \\ \gg \gg \underset{x\to 1}{\lim }\frac{(x-1{)}^2}{(x-1{)}^2}=1\end{array}$$