L'Hôpital's Rule: Conditions, Applications, and Indeterminate Forms

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What is an indeterminate form?

A limit form, such as $0/0$ or $\infty/\infty$, that does not by itself determine the limit.

State the basic L'Hôpital rule.

For an eligible $0/0$ or $\infty/\infty$ limit, if the hypotheses hold and $\lim f'(x)/g'(x)$ exists, then $\lim f(x)/g(x)=\lim f'(x)/g'(x)$.

How does L'Hôpital's rule differentiate a quotient?

It differentiates the numerator and denominator separately; it does not apply the quotient rule to $f/g$.

Why can quotients with the form $0/0$ have different limits?

The quotient depends on the relative rates at which its numerator and denominator approach zero.

What should be checked before applying L'Hôpital's rule?

Confirm an eligible indeterminate form and verify the rule's hypotheses.

Quiz questions

1. Suppose $f(x),g(x)\to 0$ as $x\to c$, the hypotheses of L'Hôpital's rule hold, and $\lim_{x\to c} f'(x)/g'(x)=L$. What follows?Answer: $\lim_{x\to c} f(x)/g(x)=L$.
2. For a positive integer $n$, what is $\lim_{x\to\infty}x^n/e^x$?Answer: $0$
3. As $x\to0^+$, how can the product $x\ln x$ be rewritten as a quotient with an eligible form?Answer: $\frac{\ln x}{1/x}$
4. For the parametric curve $(g(t),f(t))$, what is the tangent slope when $g'(t)\ne 0$?Answer: $f'(t)/g'(t)$
5. What is $\lim_{x\to0^+}x^x$, evaluated by taking logarithms?Answer: $1$