第4章微分の応用

In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.

L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider

If $\underset{x\to a}{\text{lim}}f\left(x\right)=L_1\text{and}\underset{x\to a}{\text{lim}}g\left(x\right)=L_2\ne 0,$ then

However, what happens if $\underset{x\to a}{\text{lim}}f\left(x\right)=0$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=0?$ We call this one of the indeterminate forms, of type $\frac{0}{0}.$ This is considered an indeterminate form because we cannot determine the exact behavior of $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to a$ without further analysis. We have seen examples of this earlier in the text. For example, consider

For the first of these examples, we can evaluate the limit by factoring the numerator and writing

For $\underset{x\to 0}{\text{lim}}\frac{\text{sin}x}{x}$ we were able to show, using a geometric argument, that

Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously.

The idea behind L’Hôpital’s rule can be explained using local linear approximations. Consider two differentiable functions $f$ and $g$ such that $\underset{x\to a}{\text{lim}}f\left(x\right)=0=\underset{x\to a}{\text{lim}}g\left(x\right)$ and such that $g^{\prime }\left(a\right)\ne 0$ For $x$ near $a,$ we can write

and

Therefore,

Since $f$ is differentiable at $a,$ then $f$ is continuous at $a,$ and therefore $f\left(a\right)=\underset{x\to a}{\text{lim}}f\left(x\right)=0.$ Similarly, $g\left(a\right)=\underset{x\to a}{\text{lim}}g\left(x\right)=0.$ If we also assume that $f^{\prime }$ and $g^{\prime }$ are continuous at $x=a,$ then $f^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}{f}^{\prime }\left(x\right)$ and $g^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}{g}^{\prime }\left(x\right).$ Using these ideas, we conclude that

Note that the assumption that $f^{\prime }$ and $g^{\prime }$ are continuous at $a$ and $g^{\prime }\left(a\right)\ne 0$ can be loosened. We state L’Hôpital’s rule formally for the indeterminate form $\frac{0}{0}.$ Also note that the notation $\frac{0}{0}$ does not mean we are actually dividing zero by zero. Rather, we are using the notation $\frac{0}{0}$ to represent a quotient of limits, each of which is zero.

We can also use L’Hôpital’s rule to evaluate limits of quotients $\frac{f\left(x\right)}{g\left(x\right)}$ in which $f\left(x\right)\to \text{±}\infty$ and $g\left(x\right)\to \text{±}\infty .$ Limits of this form are classified as indeterminate forms of type $\infty \text{/}\infty .$ Again, note that we are not actually dividing $\infty$ by $\infty .$ Since $\infty$ is not a real number, that is impossible; rather, $\infty \text{/}\infty .$ is used to represent a quotient of limits, each of which is $\infty$ or $\text{−}\infty .$

As mentioned, L’Hôpital’s rule is an extremely useful tool for evaluating limits. It is important to remember, however, that to apply L’Hôpital’s rule to a quotient $\frac{f\left(x\right)}{g\left(x\right)},$ it is essential that the limit of $\frac{f\left(x\right)}{g\left(x\right)}$ be of the form $\frac{0}{0}$ or $\infty \text{/}\infty .$ Consider the following example.