L'Hopital's Rule flashcards for limits practice

Question 1

What does it mean for a limit to have the indeterminate form $\frac{0}{0}$?

Accepted Answer

It means $\lim f(x)=0$ and $\lim g(x)=0$ (for the same $x	o a$ or $x	o\infty$), so $\lim \frac{f(x)}{g(x)}$ cannot be determined from substitution alone.

Question 2

Evaluate $\lim_{x	o 0}\frac{\sin x}{x}$ using L'Hopital's Rule.

Accepted Answer

$$\lim_{x	o 0}\frac{\sin x}{x}=\lim_{x	o 0}\frac{\cos x}{1}=1.$$

Question 3

Evaluate $\lim_{x	o 0}\frac{1-\cos x}{x^2}$ using repeated L'Hopital's Rule.

Accepted Answer

First gives $\frac{\sin x}{2x}$ (still $0/0$), second gives $$\lim_{x	o 0}\frac{\cos x}{2}=\frac{1}{2}.$$

L'Hopital's Rule flashcards for limits practice

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