4.4 L’Hôpital’s Rule

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  Recognize when to apply L’Hôpital’s rule.

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  Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.

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  Describe the relative growth rates of functions.

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

However, what happens if $\underset{x\to a}{lim}f(x)=0$ and $\underset{x\to a}{lim}g(x)=0\mathrm{?}$ We call this one of the indeterminate forms, of type ${\displaystyle \frac{0}{0}}$. This is considered an indeterminate form because we cannot determine the exact behavior of ${\displaystyle \frac{f(x)}{g(x)}}$ 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 $\underset{x\to 0}{lim}{\displaystyle \frac{\sin x}{x}}$ 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 $\underset{x\to a}{lim}f(x)=0=\underset{x\to a}{lim}g(x)$ and such that $g^{\prime }(a)\ne 0$. For $x$ near $a$, we can write

and

Therefore,

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

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

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

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 ${\displaystyle \frac{f(x)}{g(x)}}$, it is essential that the limit of ${\displaystyle \frac{f(x)}{g(x)}}$ be of the form ${\displaystyle \frac{0}{0}}$ or $\mathrm{\infty }/\mathrm{\infty }$. Consider the following example.

L’Hôpital’s rule is very useful for evaluating limits involving the indeterminate forms ${\displaystyle \frac{0}{0}}$ and $\mathrm{\infty }/\mathrm{\infty }$. However, we can also use L’Hôpital’s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions $0\cdot \mathrm{\infty }$, $\mathrm{\infty }-\mathrm{\infty }$, $1^{\mathrm{\infty }}$, ${\mathrm{\infty }}^0$, and $0^0$ are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits. Next we realize why these are indeterminate forms and then understand how to use L’Hôpital’s rule in these cases. The key idea is that we must rewrite the indeterminate forms in such a way that we arrive at the indeterminate form ${\displaystyle \frac{0}{0}}$ or $\mathrm{\infty }/\mathrm{\infty }$.

Suppose we want to evaluate $\underset{x\to a}{lim}\left(f(x)\cdot g(x)\right)$, where $f(x)\to 0$ and $g(x)\to \mathrm{\infty }$ (or $-\mathrm{\infty })$ as $x\to a$. Since one term in the product is approaching zero but the other term is becoming arbitrarily large (in magnitude), anything can happen to the product. We use the notation $0\cdot \mathrm{\infty }$ to denote the form that arises in this situation. The expression $0\cdot \mathrm{\infty }$ is considered indeterminate because we cannot determine without further analysis the exact behavior of the product $f(x)g(x)$ as $x\to a$. For example, let $n$ be a positive integer and consider

As $x\to \mathrm{\infty }$, $f(x)\to 0$ and $g(x)\to \mathrm{\infty }$. However, the limit as $x\to \mathrm{\infty }$ of $f(x)g(x)={\displaystyle \frac{3x^2}{(x^n+1)}}$ varies, depending on $n$. If $n=2$, then $\underset{x\to \mathrm{\infty }}{lim}f(x)g(x)=3$. If $n=1$, then $\underset{x\to \mathrm{\infty }}{lim}f(x)g(x)=\mathrm{\infty }$. If $n=3$, then $\underset{x\to \mathrm{\infty }}{lim}f(x)g(x)=0$. Here we consider another limit involving the indeterminate form $0\cdot \mathrm{\infty }$ and show how to rewrite the function as a quotient to use L’Hôpital’s rule.

Another type of indeterminate form is $\mathrm{\infty }-\mathrm{\infty }$. Consider the following example. Let $n$ be a positive integer and let $f(x)=3x^n$ and $g(x)=3x^2+5$. As $x\to \mathrm{\infty }$, $f(x)\to \mathrm{\infty }$ and $g(x)\to \mathrm{\infty }$. We are interested in $\underset{x\to \mathrm{\infty }}{lim}\left(f(x)-g(x)\right)$. Depending on whether $f(x)$ grows faster, $g(x)$ grows faster, or they grow at the same rate, as we see next, anything can happen in this limit. Since $f(x)\to \mathrm{\infty }$ and $g(x)\to \mathrm{\infty }$, we write $\mathrm{\infty }-\mathrm{\infty }$ to denote the form of this limit. As with our other indeterminate forms, $\mathrm{\infty }-\mathrm{\infty }$ has no meaning on its own and we must do more analysis to determine the value of the limit. For example, suppose the exponent $n$ in the function $f(x)=3x^n$ is $n=3$, then

On the other hand, if $n=2$, then

However, if $n=1$, then

Therefore, the limit cannot be determined by considering only $\mathrm{\infty }-\mathrm{\infty }$. Next we see how to rewrite an expression involving the indeterminate form $\mathrm{\infty }-\mathrm{\infty }$ as a fraction to apply L’Hôpital’s rule.

Another type of indeterminate form that arises when evaluating limits involves exponents. The expressions $0^0$, ${\mathrm{\infty }}^0$, and $1^{\mathrm{\infty }}$ are all indeterminate forms. On their own, these expressions are meaningless because we cannot actually evaluate these expressions as we would evaluate an expression involving real numbers. Rather, these expressions represent forms that arise when finding limits. Now we examine how L’Hôpital’s rule can be used to evaluate limits involving these indeterminate forms.

Since L’Hôpital’s rule applies to quotients, we use the natural logarithm function and its properties to reduce a problem evaluating a limit involving exponents to a related problem involving a limit of a quotient. For example, suppose we want to evaluate $\underset{x\to a}{lim}f{(x)}^{g(x)}$ and we arrive at the indeterminate form ${\mathrm{\infty }}^0$. (The indeterminate forms $0^0$ and $1^{\mathrm{\infty }}$ can be handled similarly.) We proceed as follows. Let

Then,

Therefore,

Since $\underset{x\to a}{lim}f(x)=\mathrm{\infty }$, we know that $\underset{x\to a}{lim}\ln (f(x))=\mathrm{\infty }$. Therefore, $\underset{x\to a}{lim}g(x)\ln (f(x))$ is of the indeterminate form $0\cdot \mathrm{\infty }$, and we can use the techniques discussed earlier to rewrite the expression $g(x)\ln (f(x))$ in a form so that we can apply L’Hôpital’s rule. Suppose $\underset{x\to a}{lim}g(x)\ln (f(x))=L$, where $L$ may be $\mathrm{\infty }$ or $-\mathrm{\infty }$. Then

Since the natural logarithm function is continuous, we conclude that

which gives us

Suppose the functions $f$ and $g$ both approach infinity as $x\to \mathrm{\infty }$. Although the values of both functions become arbitrarily large as the values of $x$ become sufficiently large, sometimes one function is growing more quickly than the other. For example, $f(x)=x^2$ and $g(x)=x^3$ both approach infinity as $x\to \mathrm{\infty }$. However, as shown in the following table, the values of $x^3$ are growing much faster than the values of $x^2$.

In fact,

As a result, we say $x^3$ is growing more rapidly than $x^2$ as $x\to \mathrm{\infty }$. On the other hand, for $f(x)=x^2$ and $g(x)=3x^2+4x+1$, although the values of $g(x)$ are always greater than the values of $f(x)$ for $x>0$, each value of $g(x)$ is roughly three times the corresponding value of $f(x)$ as $x\to \mathrm{\infty }$, as shown in the following table. In fact,

In this case, we say that $x^2$ and $3x^2+4x+1$ are growing at the same rate as $x\to \mathrm{\infty }$.

More generally, suppose $f$ and $g$ are two functions that approach infinity as $x\to \mathrm{\infty }$. We say $g$ grows more rapidly than $f$ as $x\to \mathrm{\infty }$ if

On the other hand, if there exists a constant $M\ne 0$ such that

we say $f$ and $g$ grow at the same rate as $x\to \mathrm{\infty }$.

Next we see how to use L’Hôpital’s rule to compare the growth rates of power, exponential, and logarithmic functions.