Question

Evaluate the limit. Evaluate the limit \(\lim _{x \rightarrow \infty} \frac{e^{x}}{x}\).

Short answer

The limit is infinity as \( x \rightarrow \infty \).

Step-by-step solution

Identify the Limit Type

Recognize that the expression \( \frac{e^{x}}{x} \) involves a limit at \( x \rightarrow \infty \). This is an indeterminate form \( \frac{\infty}{\infty} \), so further analysis is needed to determine the limit.

Apply L'Hôpital's Rule

Since the limit \( \lim_{x \rightarrow \infty} \frac{e^{x}}{x} \) is in an indeterminate form \( \frac{\infty}{\infty} \), L'Hôpital's Rule can be used. According to this rule, if \( \lim_{x \rightarrow c} \frac{f(x)}{g(x)} \) is indeterminate, then \( \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)} \), provided the limit on the right exists.

Differentiate the Numerator and Denominator

Take the derivative of the numerator and the denominator: - The derivative of \( e^{x} \) is \( e^{x} \).- The derivative of \( x \) is \( 1 \).Thus, the new expression is \( \lim_{x \rightarrow \infty} \frac{e^{x}}{1} \).

Evaluate the New Limit

Now evaluate the new limit: \( \lim_{x \rightarrow \infty} e^{x} \), which is \( \infty \), because \( e^x \) grows exponentially as \( x \) increases.

Conclusion

The limit \( \lim_{x \rightarrow \infty} \frac{e^{x}}{x} \) is equal to \( \infty \). Therefore, as \( x \) approaches infinity, \( \frac{e^{x}}{x} \) grows without bound.

Key concepts

Indeterminate Forms

When evaluating limits, particularly as a variable approaches a value like infinity, indeterminate forms often arise. These forms are expressions where the limit isn't immediately clear, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In the problem provided, \( \lim_{x \rightarrow \infty} \frac{e^{x}}{x} \) is a classic example of an indeterminate form \( \frac{\infty}{\infty} \).
To resolve such cases, we need special techniques since substituting values into these forms doesn't yield a definitive answer. L'Hôpital's Rule, for instance, gives us a way to determine limits that initially seem undecidable. This rule specifically applies to indeterminate forms, and by differentiating the numerator and denominator separately, you may find a limit that actually exists.

Exponential Functions

The function \( e^x \) is a fundamental example of exponential growth. Exponential functions are characterized by a constant base raised to a variable exponent, and they have important properties that make them grow very quickly.
- When \( x \) is large, \( e^x \) becomes significantly bigger than polynomial expressions such as \( x \). - They appear frequently in real-world contexts like compound interest, population growth, and radioactive decay. - One interesting property is that the derivative of \( e^x \) is also \( e^x \), which reflects its rapidly increasing nature.
Recognizing this same function in the numerator of \( \frac{e^{x}}{x} \) helps understand why in limits the exponential growth overpowers the linear growth of \( x \), leading to an indeterminate form which can be solved using L'Hôpital's Rule.

Limits at Infinity

Limits at infinity refer to the behavior of a function as its input becomes very large (positively or negatively). It's about predicting the long-term trend of a curve as it heads towards infinity.
In the exercise \( \lim_{x \rightarrow \infty} \frac{e^{x}}{x} \), you're exploring how the function behaves as \( x \) grows without bound. Usually: - If a numerator grows faster than the denominator (as is the case with \( e^x \) compared to \( x \)), the overall limit tends to infinity. - Conversely, if the denominator grows faster, the limit might approach zero instead.
Understanding infinity in limits helps in visualizing and determining end behavior in various functions and is crucial for comprehending advanced calculus concepts. It helps in transforming difficult expressions into manageable solutions using methodical approaches such as L'Hôpital's Rule.