Problem 386 - Calculus Volume 1

Question

For the following exercises, evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow \infty}\left(x-e^{x}\right) $$

Step-by-Step Solution

Answer

The limit evaluates to \(-\infty\).

Step 1: Determine the Form of the Limit

As \( x \to \infty \), we have \( x - e^x \). We need to determine whether this is an indeterminate form or not. Substitute some large values mentally: \( e^x \) grows much faster than \( x \), implying \( x - e^x \) approaches \(-\infty\). However, to apply L'Hôpital's Rule, we need the form \( \frac{f(x)}{g(x)} \). Initially, the form presented is not obviously suitable for L'Hôpital's Rule.

Step 2: Rewrite the Limit for L'Hôpital's Rule

Rewrite \( x - e^x \) in a form that is suitable for L'Hôpital's Rule. Consider \( \lim _{x \to \infty} \frac{x}{e^x} \). This rewrite aims to convert a subtraction into a division, which might result in an indeterminate form \( \frac{\infty}{\infty} \).

Step 3: Apply L'Hôpital's Rule

Since \( \lim _{x \to \infty} \frac{x}{e^x} \) is in the indeterminate form \( \frac{\infty}{\infty} \), we apply L'Hôpital's Rule by differentiating the numerator and the denominator:\[ \lim _{x \to \infty} \frac{x}{e^x} = \lim _{x \to \infty} \frac{1}{e^x} \]Differentiate \( x \) to get \( 1 \) and \( e^x \) to get \( e^x \).

Step 4: Evaluate the Limit

Now solve the limit after applying L'Hôpital's Rule:\[ \lim _{x \to \infty} \frac{1}{e^x} = 0 \]As \( x \to \infty \), \( e^x \to \infty \), making \( \frac{1}{e^x} \to 0 \).

Step 5: Conclusion

Therefore, the original limit \( \lim_{x \to \infty} (x - e^x) = -\infty \), but by using the division approach and L'Hôpital's Rule, we confirmed that the rate of growth leads \( x \) to become irrelevant compared to \( e^x \), leading to the final answer. The manipulation allowed us to focus on how the terms approach their asymptotic behavior.

Key Concepts

Indeterminate Forms

When calculating limits, especially as variables approach infinity or zero, we sometimes encounter indeterminate forms. Understanding these forms is crucial for determining the appropriate mathematical techniques to solve a problem, such as those involving L'Hôpital's Rule.Indeterminate forms arise when substituting the limit directly into an expression results in forms like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or \( \infty - \infty \). These forms do not provide clear information about the behavior of the function as it approaches a given limit.In the exercise provided, the expression \(x - e^x\) as \(x\) approaches infinity results in a situation that appears to approach \(-\infty\). However, for using L'Hôpital's Rule effectively, converting the expression into a division form like \( \frac{f(x)}{g(x)} = \frac{\infty}{\infty} \) is more appropriate.This allows for the application of derivatives on the numerator and the denominator, simplifying the limit evaluation.

Limits at Infinity

Limits at infinity help describe the behavior of functions as the input value approaches positive or negative infinity. This concept is significant in understanding the end behavior of functions, especially in calculus and analysis. When evaluating limits at infinity, it's essential to consider how different parts of the function grow relative to each other, impacting the overall limit.In our example, \( \lim_{x \to \infty} (x - e^x) \), we look at how the linear part \(x\) and the exponential part \(e^x\) behave. Over large values, \(e^x\) grows significantly faster than \(x\), which suggests that \(x\) becomes trivial in comparison, leading the difference to negative infinity.By translating it into \( \lim_{x \to \infty} \frac{x}{e^x} \), the indeterminate form \( \frac{\infty}{\infty} \) appears, allowing us to use L'Hôpital's Rule. This highlights how limits can be transformed to enable simpler calculus techniques for evaluation.

Exponential Functions

Exponential functions, such as \(e^x\), play a crucial role in calculus and analysis due to their unique rate of growth. They grow faster than polynomial functions and have applications across various scientific fields, making them a vital part of mathematical study.The defining trait of an exponential function is that its rate of increase grows proportionally with its current value. For example, \(e^x\) grows much more rapidly than \(x\), even for large values of \(x\). This distinct property sets exponential functions apart when evaluating limits, especially at infinity.In the exercise, \(e^x\) dominates any linear terms, meaning that as \(x\) approaches infinity, the impact of \(x\) on the expression \(x - e^x\) diminishes, resulting in the expression tending towards \(-\infty\). Thus, understanding exponential growth provides insight into solving such limits using L'Hôpital's Rule or other methods.