L'Hôpital's Rule

L’Hôpital’s rule is used to evaluate limits of the indeterminate form $\frac{\infty }{\infty }$, as long as both the numerator and denominator functions are differentiable. It’s more tedious, so check if common terms can be factored out or cancelled.

Definition §

Let $f,g$ be real-valued functions.

${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$

• The function must be in the form $\frac{\infty }{\infty }$ or $\frac{0}{0}$ .
• Given an open interval ((3,4), (a,b), etc.) $D$ where $x\ne a$:
  • $f$ and $g$ must be differentiable in $D$
  • $g^{\prime }(x)\ne 0$ in $D$
• Obviously, ${\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ needs to exist

The rule also works for one-sided limits.

Uses In Proving Limits §

L’Hôpital’s rule can be used to prove a limit exists, but it cannot be used to show that a limit does not exist. I.e. there are functions which the rule does not work for, but the limit does exist:

Counterexample

Find ${\lim }_{x\to \infty }\frac{x+\sin (x)}{x}$

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