Problem Set - L'Hôpital's Rule

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

L'Hôpital's Rule

	State L'Hôpital's rule and give an example which shows how it is used.
	Explain why L'Hôpital's rule works. Hint: Expand the numerator and the denominator in terms of $\Delta x$.
	Give three examples which illustrate that a limit problem that looks like it is coming out to $0/0$ could really be getting closer and closer to almost anything and must be looked at in a different way.
	Give three examples which illustrate that a limit problem that looks like it is coming out to $1^{\infty }$ could really be getting closer and closer to almost anything and must be looked at in a different way.
	Give three examples which illustrate that a limit problem that looks like it is coming out to $0^0$ could really be getting closer and closer to almost anything and must be looked at in a different way.
	Evaluate ${\displaystyle \underset{x\to 1}{\lim }\frac{x^2+3x-4}{x-1}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{\lim }\frac{x^a-1}{x^b-1}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{\lim }\frac{\ln x}{x-1}}$.
	Evaluate ${\displaystyle \underset{x\to \pi }{\lim }\frac{\tan x}{x-\pi }}$.
	Evaluate ${\displaystyle \underset{x\to 3\pi /2}{\lim }\frac{\cos x}{x-\left(3\pi /2\right)}}$.
	Evaluate ${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{\ln x}{\sqrt{x}}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }\frac{{\left(\ln x\right)}^3}{x^2}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{6^x-2^x}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{e^x-1-x-\left(x^2/2\right)}{x^3}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{\sin x-x}{x^3}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }\frac{\ln \left(1+e^x\right)}{5x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{\tan \alpha x}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{2x-\arcsin x}{2x-\arccos x}}$.
	Evaluate ${\displaystyle \underset{x\to 0^{+}}{\lim }\sqrt{x} \ln x}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }{e}^{-x} \ln x}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }{x}^3{e}^{-x^2}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }\left(x-\pi \right)\cot x}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }{x}^{-4}-x^{-2}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }{x}^{-1}-\csc x}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }x-\sqrt{x^2-1}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }\left(\frac{x^3}{x^2-1}-\frac{x^3}{x^2+1}\right)}$.
	Evaluate ${\displaystyle \underset{x\to 0^{+}}{\lim }{x}^{\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }{\left(1-2x\right)}^{1/x}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }{\left(1+3/x+5/x^2\right)}^x}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }{x}^{1/x}}$.
	Evaluate ${\displaystyle \underset{x\to 0^{+}}{\lim }{\left(\cot x\right)}^{\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{\lim }{\left(\frac{x}{x-1}\right)}^x}$.
	Evaluate ${\displaystyle \underset{x\to 0^{+}}{\lim }{\left(-\ln x\right)}^x}$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)