Indeterminate Forms

From the results of the previous chapters, we do not know how to find lim x → x 0 f ( x ) / g ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351180641/0424c007-6610-4c93-8ae3-088a74bd8b1d/content/eq1472.tif"/> when lim x → x 0 f ( x ) = lim x → x 0 g ( x ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351180641/0424c007-6610-4c93-8ae3-088a74bd8b1d/content/eq1473.tif"/> . In this chapter, we will discuss L’Hôpital’s rule (originally due to John Bernoulli) which enables us to determine limits of functions that are not only in an indeterminate form 0/0, but also ∞/∞, and perhaps even ∞ − ∞, 0 · ∞, 0⁰, ∞⁰, and 1^∞. Notice that two forms of L’Hôpital’s rule for sequences have already been discussed in Theorem 8.4 and Problem 8.15.