Indeterminate Forms and L'Hopitals Rule
Indeterminate Forms and L'Hopitals Rule
Objectives
- Recognize when to apply L’Hôpital’s rule.
- Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
- Describe the relative growth rates of functions.
Summary
Derivatives can be used to help us evaluate indeterminate limits of the form \(\frac{0}{0}\) through L'Hôpital's Rule, by replacing the functions in the numerator and denominator with their tangent line approximations. In particular, if \(f(a) = g(a) = 0\) and \(f\) and \(g\) are differentiable at \(a\text{,}\) L'Hôpital's Rule tells us that
\begin{equation*} \lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}\text{.} \end{equation*}When we write \(x \to \infty\text{,}\) this means that \(x\) is increasing without bound. Thus, \(\lim_{x \to \infty} f(x) = L\) means that we can make \(f(x)\) as close to \(L\) as we like by choosing \(x\) to be sufficiently large. Similarly, \(\lim_{x \to a} f(x) = \infty\text{,}\) means that we can make \(f(x)\) as large as we like by choosing \(x\) sufficiently close to \(a\text{.}\)
A version of L'Hôpital's Rule also helps us evaluate indeterminate limits of the form \(\frac{\infty}{\infty}\text{.}\) If \(f\) and \(g\) are differentiable and both approach zero or both approach \(\pm \infty\) as \(x \to a\) (where \(a\) is allowed to be \(\infty\)), then
\begin{equation*} \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\text{.} \end{equation*}