Problem 83 - Calculus Early Transcendentals

Question

Explain why l'Hôpital's Rule fails when applied to the limit \(\lim _{x \rightarrow \infty} \frac{\sinh x}{\cosh x}\) and then find the limit another way.

Step-by-Step Solution

Answer

Answer: L'Hôpital's Rule does not apply in this case because applying it results in an infinite cycle of indeterminate forms. The correct limit is 1, which can be found by using the properties of hyperbolic functions and simplifying the expression.

Step 1: Understanding l'Hôpital's Rule

L'Hôpital's Rule can be used to find the limit of a ratio of two functions when the limit yields an indeterminate form of \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). It states that if \(\lim_{x\to c} \frac{f'(x)}{g'(x)}\) exists, then \(\lim_{x\to c} \frac{f(x)}{g(x)}\) also exists and has the same value.

Step 2: Evaluating the given limit using l'Hôpital's Rule

In order to apply l'Hôpital's Rule, we first need to check if the given limit \(\lim _{x\rightarrow \infty} \frac{\sinh x}{\cosh x}\) is an indeterminate form of \(\frac{\infty}{\infty}\). As \(x\) approaches infinity, both \(\sinh(x)\) and \(\cosh(x)\) go to infinity, so we do in fact have an indeterminate form of \(\frac{\infty}{\infty}\). Now, let's apply l'Hôpital's Rule by differentiating the numerator and denominator: The derivative of \(\sinh(x)\) is \(\cosh(x)\), and the derivative of \(\cosh(x)\) is \(\sinh(x)\). So, we get: $$ \lim _{x\rightarrow \infty} \frac{\cosh(x)}{\sinh(x)} $$ This limit is still in the indeterminate form of \(\frac{\infty}{\infty}\). If we continue to apply l'Hôpital's Rule, we will get an infinite cycle, which means l'Hôpital's Rule fails for this limit.

Step 3: Finding the limit using an alternative method

We will find the limit using the properties of hyperbolic functions. Recall that \(\sinh(x) = \frac{e^x-e^{-x}}{2}\) and \(\cosh(x) = \frac{e^x+e^{-x}}{2}\). Thus, we can rewrite the limit as: $$ \lim _{x\rightarrow \infty} \frac{\frac{e^x-e^{-x}}{2}}{\frac{e^x+e^{-x}}{2}} $$ Dividing the numerator and the denominator by the common factor, \(e^x\), we obtain: $$ \lim _{x\rightarrow \infty} \frac{1-e^{-2x}}{1+e^{-2x}} $$ Now, as \(x\) approaches infinity, \(e^{-2x}\) approaches 0. So the limit becomes: $$ \lim _{x\rightarrow \infty} \frac{1-0}{1+0} = \frac{1}{1} = 1 $$ Thus, the limit of the given function is 1.

Key Concepts

Limits in Calculus

In calculus, limits are fundamental for understanding how functions behave as their inputs get very close to some value, including infinity. This concept is pivotal for various parts of calculus, such as the derivative, which is itself defined as a limit.

When we look at the expression \(\lim_{x\to c} f(x)\), it represents the value that the function \(f(x)\) approaches as \(x\) approaches the value \(c\). There can, however, be cases where this limit is not straightforward to calculate, especially when it brings us to an indeterminate form. For such cases, advanced techniques, like l'Hôpital's Rule, are employed to facilitate easier computation of these limits.

Hyperbolic Functions

Hyperbolic functions, which include \(\sinh(x)\) and \(\cosh(x)\), have important applications in many areas of mathematics, including calculus. They are analogs of the trigonometric functions but for the hyperbola, just like sine and cosine are for the circle.

The hyperbolic sine function, \(\sinh(x)\), is defined as \(\sinh(x) = \frac{e^x-e^{-x}}{2}\), while the hyperbolic cosine, \(\cosh(x)\), is \(\cosh(x) = \frac{e^x+e^{-x}}{2}\). These functions grow exponentially and are crucial when dealing with the limits involving growth and decay phenomena in various real-world contexts.

Indeterminate Forms

Indeterminate forms arise in limit problems where the limit expression does not clearly lead to a specific answer. Such forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0\cdot\infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), and \(1^\infty\).

These expressions do not have an immediate determinate value, and special methods such as l'Hôpital's Rule are used to find the limits that result in these forms. It's important to recognize these forms because applying the standard limit rules or intuition directly often leads to incorrect conclusions.

Differentiation

Differentiation is a process in calculus which measures how a function changes as its input changes. The derivative of a function at a certain point is the slope of the tangent line to the function's graph at that point. It symbolizes the rate at which the function value is changing at that particular point in terms of its input value.

The process of differentiation is applied in l'Hôpital's Rule to resolve indeterminate forms by differentiating both the numerator and the denominator of a function separately. For instance, the differentiation of \(\sinh(x)\) results in \(\cosh(x)\), and vice versa, which creates a cycle if attempting l'Hôpital's Rule on \(\frac{\sinh x}{\cosh x}\) as we approach infinity.