Limits in indeterminate forms (HL)

This expands upon the discussion in Limits, continuity and differentiability. Here we elaborate on

Contents

• Indeterminate forms
• L’Hôpital’s rule
• Extension
• Alternative method

Indeterminate forms

The indeterminate forms are what that would result, if the single limit were split into two. They are

Limits in indeterminate forms may evaluate to any real number, as well as undefined. The l’Hôpital’s rule, possibly through repeated use, can determine the value of some of such limits.

L’Hôpital’s rule

The l’Hôpital’s rule can be used on $\displaystyle \lim_{x\to c}\frac{f(x)}{g(x)}$ when all of the following are true

1. The fraction is in the $\frac00$ or $\pm \frac\infty\infty$ form. (As we’ll briefly touch upon, other indeterminate forms can be written as one of these two, though only these two are tested in IB Math AA.)
2. The limit $\displaystyle \lim_{x\to c}\frac{f^\prime(x)}{g^\prime(x)}$ exists, which additionally requires both $f(x)$ and $g(x)$ to be differentiable near $x = c$.

Then

L’Hôpital’s rule may be applied multiple times, as long as each intermediate result is indeterminate.

When the question in IB Math AA explicitly mentions l’Hôpital’s rule, then the limit exists. Though, there are cases when $\displaystyle \lim_{x\to c}\frac{f^\prime(x)}{g^\prime(x)}$ does not exist, but $\displaystyle \lim_{x\to c}\frac{f(x)}{g(x)}$ does, or vice versa. For examples of when these may occur, refer to the wikipedia article 

Extension

These are technically not part of the syllabus, but they can be useful to know

For $f(x) \cdot g(x)$ in indeterminate form, use

Which one goes on the denominator is entirely personal taste, depending on what is easier to differentiate.

For $f(x)^{g(x)}$ in indeterminate form

Finally, $\infty -\infty$ is the trickiest but they may have a common factor or it’s possible to convert two fractions into one with a common denominator.

Alternative method

Many limits can be expressed using Maclaurin series, which transforms the limit to a limit of a polynomial. See the bottom of the Maclaurin series page for a practical example.