Active Calculus

The idea demonstrated in Preview Activity 2.8.1 — that we can evaluate an indeterminate limit of the form \(\frac{0}{0}\) by replacing each of the numerator and denominator with their local linearizations at the point of interest — can be generalized in a way that enables us to evaluate a wide range of limits. We have a function \(h(x)\) that can be written as a quotient \(h(x) = \frac{f(x)}{g(x)}\text{,}\) where \(f\) and \(g\) are both differentiable at \(x=a\) and for which \(f(a) = g(a) = 0\text{.}\) We would like to evaluate the indeterminate limit given by \(\lim_{x \to a} h(x)\text{.}\) Figure 2.8.1 illustrates the situation. We see that both \(f\) and \(g\) have an \(x\)-intercept at \(x = a\text{.}\) Their respective tangent line approximations \(L_f\) and \(L_g\) at \(x = a\) are also shown in the figure. We can take advantage of the fact that a function and its tangent line approximation become indistinguishable as \(x \to a\text{.}\)

This result holds as long as \(g'(a)\) is not equal to zero. The formal name of the result is L’Hôpital’s Rule.

By replacing the numerator and denominator with their respective derivatives, we often replace an indeterminate limit with one whose value we can easily determine.

While L’Hôpital’s Rule can be applied in an entirely algebraic way, it is important to remember that the justification of the rule is graphical: the main idea is that the slopes of the tangent lines to \(f\) and \(g\) at \(x = a\) determine the value of the limit of \(\frac{f(x)}{g(x)}\) as \(x \to a\text{.}\)

It’s not the fact that \(f\) and \(g\) both approach zero that matters most, but rather the rate at which each approaches zero that determines the value of the limit. This is a good way to remember what L’Hôpital’s Rule says: if \(f(a) = g(a) = 0\text{,}\) the the limit of \(\frac{f(x)}{g(x)}\) as \(x \to a\) is given by the ratio of the slopes of \(f\) and \(g\) at \(x = a\text{.}\)

The concept of infinity, denoted \(\infty\text{,}\) arises naturally in calculus, as it does in much of mathematics. It is important to note from the outset that \(\infty\) is a concept, but not a number itself. Indeed, the notion of \(\infty\) naturally invokes the idea of limits. Consider, for example, the function \(f(x) = \frac{1}{x}\text{,}\) whose graph is pictured in Figure 2.8.4.

We note that \(x = 0\) is not in the domain of \(f\text{,}\) so we may naturally wonder what happens as \(x \to 0\text{.}\) As \(x \to 0^+\text{,}\) we observe that \(f(x)\) increases without bound. That is, we can make the value of \(f(x)\) as large as we like by taking \(x\) closer and closer (but not equal) to 0, while keeping \(x \gt 0\text{.}\) This is a good way to think about what infinity represents: a quantity is tending to infinity if there is no single number that the quantity is always less than.

Limits involving infinity identify vertical and horizontal asymptotes of a function. If \(\lim_{x \to a} f(x) = \infty\text{,}\) then \(x = a\) is a vertical asymptote of \(f\text{,}\) while if \(\lim_{x \to \infty} f(x) = L\text{,}\) then \(y = L\) is a horizontal asymptote of \(f\text{.}\) Similar statements can be made using \(-\infty\text{,}\) and with left- and right-hand limits as \(x \to a^-\) or \(x \to a^+\text{.}\)

In precalculus classes, it is common to study the end behavior of certain families of functions, by which we mean the behavior of a function as \(x \to \infty\) and as \(x \to -\infty\text{.}\) Here we briefly examine some familiar functions and note the values of several limits involving \(\infty\text{.}\)

For the natural exponential function \(e^x\text{,}\) we note that \(\lim_{x \to \infty} e^x = \infty\) and \(\lim_{x \to -\infty} e^x = 0\text{.}\) For the exponential decay function \(e^{-x}\text{,}\) these limits are reversed, with \(\lim_{x \to \infty} e^{-x} = 0\) and \(\lim_{x \to -\infty} e^{-x} = \infty\text{.}\) Turning to the natural logarithm function, we have \(\lim_{x \to 0^+} \ln(x) = -\infty\) and \(\lim_{x \to \infty} \ln(x) = \infty\text{.}\) While both \(e^x\) and \(\ln(x)\) grow without bound as \(x \to \infty\text{,}\) the exponential function does so much more quickly than the logarithm function does. We’ll soon use limits to quantify what we mean by “quickly.”

A function can fail to have a limit as \(x \to \infty\text{.}\) For example, consider the plot of the sine function at right in Figure 2.8.5. Because the function continues oscillating between \(-1\) and \(1\) as \(x \to \infty\text{,}\) we say that \(\lim_{x \to \infty} \sin(x)\) does not exist.

Finally, it is straightforward to analyze the behavior of any rational function as \(x \to \infty\text{.}\)

Here, both \(x^2 \to \infty\) and \(e^x \to \infty\text{,}\) but there is not an obvious algebraic approach that enables us to find the limit’s value. Fortunately, it turns out that L’Hôpital’s Rule extends to cases involving infinity.

(To be technically correct, we need to add the additional hypothesis that \(g'(x) \ne 0\) on an open interval that contains \(a\) or in every neighborhood of infinity if \(a\) is \(\infty\text{;}\) this is almost always met in practice.)