Section 12.1

L'Hôpital's rule

viz

\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)},\qquad \frac{f(x)}{g(x)}\to 0/0\text{ or }\infty/\infty,\ f,g\text{ differentiable in a punctured neighborhood of }a,\ g'(x)\neq 0\text{ on that neighborhood},\ \lim_{x\to a}\frac{f'(x)}{g'(x)}\ \text{exists (finite or }\pm\infty\text{)}

When the original limit is $0/0$ or $\infty/\infty$ and the differentiability hypotheses hold.

$\frac{f(x)}{g(x)}\to 0/0\text{ or }\infty/\infty,\ f,g\text{ differentiable in a punctured neighborhood of }a,\ g'(x)\neq 0\text{ on that neighborhood},\ \lim_{x\to a}\frac{f'(x)}{g'(x)}\ \text{exists (finite or }\pm\infty\text{)}$

\lim_{x\to\infty}\frac{f(x)}{g(x)}=\lim_{x\to\infty}\frac{f'(x)}{g'(x)},\qquad \frac{f(x)}{g(x)}\to 0/0\text{ or }\infty/\infty,\ f,g\text{ differentiable in a neighborhood of }\infty,\ g'(x)\neq 0\text{ on that neighborhood},\ \lim_{x\to\infty}\frac{f'(x)}{g'(x)}\ \text{exists (finite or }\pm\infty\text{)}

Same rule as DIF-114 at infinity; requires form $0/0$ or $\infty/\infty$ .

$\frac{f(x)}{g(x)}\to 0/0\text{ or }\infty/\infty,\ f,g\text{ differentiable in a neighborhood of }\infty,\ g'(x)\neq 0\text{ on that neighborhood},\ \lim_{x\to\infty}\frac{f'(x)}{g'(x)}\ \text{exists (finite or }\pm\infty\text{)}$