Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} e^{\cos x} $$

Step-by-Step Solution

Answer

The limit does not exist because \(e^{\cos x}\) oscillates between \(e^{-1}\) and \(e^1\) as \(x\) approaches infinity.

Step 1: Analyze the Function

First, identify the behavior of the function as \(x\) approaches infinity. The function is \(e^{\cos x}\). Since \(\cos x\) oscillates between \(-1\) and \(1\) as \(x\) increases, \(e^{\cos x}\) will oscillate between \(e^{-1}\) (which is approximately 0.3679) and \(e^{1}\) (which is approximately 2.718).

Step 2: Check for an Indeterminate Form

Before applying l'Hôpital's Rule, check if the function exhibits an indeterminate form. l'Hôpital's Rule is applicable for forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), but here, since \(e^{\cos x}\) is oscillating within a bounded range, it does not present an indeterminate form.

Step 3: Conclude the Limit

As \(x\) approaches infinity, \(e^{\cos x}\) continues to oscillate between \(e^{-1}\) and \(e^1\). Therefore, the limit does not exist in the traditional sense because it does not approach a single value.

Key Concepts

Indeterminate Forms

When studying limits, it is important to recognize situations involving indeterminate forms. Indeterminate forms occur when substituting values into a limit expression results in an undefined form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms indicate uncertainty in the value of a limit, making it unclear what the outcome should be. L'Hôpital's Rule is a useful tool that applies to specific indeterminate forms. It can simplify the evaluation of limits by differentiating the numerator and denominator separately and then taking the limit again. However, it’s crucial to confirm a function presents an indeterminate form before applying this rule. In the exercise, \(e^{\cos x}\) does not result in an indeterminate form because it doesn’t match these specific cases. As a result, l'Hôpital's Rule is not applicable.

Limit Oscillation

The concept of limit oscillation is essential to understand situations where a function does not converge to a single value. Instead of reaching a specific limit, the function continues to oscillate between values. This means the limit does not exist in the traditional sense. In the given exercise, \(e^{\cos x}\) is an example of a function that oscillates. The cosine function, \(\cos x\), oscillates between -1 and 1 as \(x\) approaches infinity. Therefore, \(e^{\cos x}\) fluctuates between \(e^{-1}\) (approximately 0.3679) and \(e^{1}\) (approximately 2.718).

• \(\cos x\) oscillates, leading \(e^{\cos x}\) to oscillate accordingly.
• Because the function keeps changing values within this range, it doesn’t settle on a single limit.
• Thus, the traditional limit is said not to exist for \(x \to \infty\).

Exponential Functions

Exponential functions are fundamental in calculus and exhibit unique properties. The general form of an exponential function is \(a^x\), where \(a\) is a constant. A common case is the natural exponential function \(e^x\), where \(e\) is approximately equal to 2.718. Exponential functions are known for their rapid growth. However, when combined with oscillating functions such as the cosine function, they can display interesting behaviors, such as oscillation.
In the exercise, \(e^{\cos x}\) showcases how exponential behavior is affected by oscillation. Since \(\cos x\) varies between -1 and 1, \(e^{\cos x}\) doesn’t simply increase or decrease. Instead, it varies within a set range.

• The base function \(e^x\) is modified by an oscillating power, \(\cos x\).
• This interplay causes \(e^{\cos x}\) to oscillate between two exponential bounds.
• The behavior is unique and indicates the range within which the exponential value will lie.

Understanding these properties of exponential functions allows one to better analyze scenarios like the one in the exercise, where oscillation impacts exponentiation.