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[h] IB Mathematics AA HL Flashcards -The evaluation of limits
[q] L’Hôpital’s Rule
L’HÔPITAL’S RULE ⟶ This rule is a very helpful method of evaluating limits of rational functions. Specifically, it will be used most commonly for limits that initially give us results such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).
[q]
The rule simply says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives:
\[
\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \quad \text{where a < c < b, if f & g are differentiable in the interval [a, b]}
\]
E.G. 2 ⟶ Find \(\lim_{{x \to ∞}} e^x \cdot x^{-x}\):
⟶ This does not seem as if it is in the correct form, but we can use the fact that \(x^{-x} = \frac{1}{x^x}\). Rewrite it as \(\lim_{{x \to ∞}} \frac{e^x}{x^x}\), then:
\[
\lim_{{x \to ∞}} \frac{e^x}{x^x} = \lim_{{x \to ∞}} \frac{e^x}{e^x \cdot x} = \frac{\infty}{1} = \infty
\]
[a]
One use of this rule is simple enough, but if they want to challenge you a little more, you might need to use it multiple times. This is when we have \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), so we use the rule, but then we still have the same problem. So we differentiate again, and see if we now have a limit that works, i.e.:
\[
\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} = \lim_{{x \to c}} \frac{f”(x)}{g”(x)} = \dots
\]
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