When can you use L'Hopital's rule

L'Hôpital's rule turns a hard indeterminate limit into a usually simpler one by differentiating the top and bottom. It is a shortcut, not a universal tool, so checking the hypotheses prevents wasted work.

Key Conditions

You must check all of these:

• Both $f$ and $g$ are differentiable on an interval around $a$ (except possibly at $a$)
• $\lim_{x\to a}f(x)=\lim_{x\to a}g(x)=0$ or both are infinite, giving 0/0 or $\infty/\infty$
• $g'(x)\neq0$ on that interval
• The limit $\lim_{x\to a}\frac{f'(x)}{g'(x)}$ exists (or is $\pm\infty$ )

Apply the rule step‑by‑step:

1. 1Confirm the original limit is 0/0 or $\infty/\infty$
2. 2Differentiate numerator and denominator separately
3. 3Take the limit of the new fraction
4. 4If the new limit is still indeterminate, repeat the process

Typical indeterminate forms and whether L'Hôpital applies

Concrete Example

Evaluate $\displaystyle\lim_{x\to0}\frac{\sin x}{x}$. Direct substitution gives 0/0, so differentiate: $\frac{d}{dx}\sin x=\cos x$ and $\frac{d}{dx}x=1$. The new limit is $\lim_{x\to0}\frac{\cos x}{1}=\cos0=1$, so the original limit equals 1.