Three Versions of L'Hospital's Rule

Here we present three versions of L'Hospital's rule with proof, which we will call Baby L'Hospital's rule, Macho L'Hospital's rule and the Extended L'Hospital's rule. The material here is more advanced than the rest of the learning module, and frequently is not required. (Check with your instructor.) The proofs on the following page are supplementary material, and are not part of the M408K/N syllabus.

• Baby L'Hospital's Rule
  
  

  Theorem: Let $f(x)$ and $g(x)$ be continuous functions on an interval containing $x=a$, with $f(a)=g(a)=0$. Suppose that $f$ and $g$ are differentiable, and that $f'$ and $g'$ are continuous. Finally, suppose that $g'(a) \ne 0$. Then $$ \lim_{x\to a}\, \frac{f(x)}{g(x)} = \lim_{x \to a} \,\frac{f'(x)}{g'(x)} = \frac{f'(a)}{g'(a)}.$$ Also, $$\lim_{x \to a^+}\, \frac{f(x)}{g(x)} = \lim_{x\to a^+} \,\frac{f'(x)}{g'(x)}$$ and $$\lim_{x \to a^-}\, \frac{f(x)}{g(x)} = \lim_{x \to a^-}\,\frac{f'(x)}{g'(x)}.$$

  
  (Proof.)

  The baby version is good enough to compute limits like \begin{equation}\lim_{x \to 0}\, \frac{\sin(2x)}{x+x^2}.\end{equation}

  However, it isn't good enough to compute limits like \begin{equation} \lim_{x \to 0} \,\frac{1 -    \cos(2x)}{x^2}, \end{equation} since in that case $g'(0)=0$.  To compute limits like that, we need the macho version.



• Macho L'Hospital's Rule:
  
  

  Theorem: Suppose that $f$ and $g$ are continuous on a closed interval $[a,b]$, and are differentiable on the open interval $(a,b)$. Suppose that $g'(x)$ is never zero on $(a,b)$, and that $\displaystyle\lim_{x \to a^+} \frac{f'(x)}{g'(x)}$ exists, and that $\displaystyle\lim_{x \to a^+} f(x) = \lim_{x\to a^+}g(x)=0$. Then $$\lim_{x \to a^+}\, \frac{f(x)}{g(x)} = \lim_{x\to a^+} \,\frac{f'(x)}{g'(x)}.$$

  
  (Proof.)

  Note that this theorem doesn't require anything about $g'(a)$, just about how $g'$ behaves to the right of $a$. An analogous theorem applies to the limit as $x \to a^-$ (and requires $f$ and $g$ and $f'$ and $g'$ to be defined on an interval that ends at $a$, rather than one that starts at $a$). You can combine the two to get a theorem about an overall limit as $x \to a$.

  The conclusion of Macho L'Hospital's Rule relates one limit (of $f/g$) to another limit (of $f'/g'$), and not to the value of $f'(a)/g'(a)$. This is what allows the theorem to be used recursively to solve problems like $\displaystyle\lim_{x \to 0} \frac{1-\cos(2x)}{x^2}$.



• Extended L'Hospital's Rule:
  
  

  Theorem: The previous versions apply to indefinite forms of type "$\infty/\infty$" as well as "$0/0$", and apply to limits as $x\to \pm \infty$ as well as to limits $x \to a^\pm$. In all of these cases, the rule is: $$ \lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}.$$

  
  (Proof.)