Question

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

Short answer

The limit is \(-\infty\).

Step-by-step solution

Identify the Form

First, we identify the form of the limit as \( x \to \infty \). The function \( x - e^x \) approaches \( \infty - \infty \) which is an indeterminate form. Therefore, we need to use additional mathematical techniques to evaluate it.

Rewrite for L'Hôpital's Rule

Rewrite the expression to use L'Hôpital's Rule. We can express the function as \( \frac{x}{e^x} - 1 \). As \( x \to \infty \), this becomes \( \frac{\infty}{\infty} - 1 \), which suggests L'Hôpital's Rule could be useful for \( \frac{x}{e^x} \).

Apply L'Hôpital's Rule

Differentiate the numerator and denominator separately: - The derivative of the numerator, \( x \), is \( 1 \).- The derivative of the denominator, \( e^x \), is \( e^x \).Therefore, \[ \frac{d}{dx}\left(\frac{x}{e^x}\right) = \frac{1}{e^x} \] Applying L'Hôpital's Rule, evaluate: \[ \lim_{x \to \infty} \frac{1}{e^x} \]

Evaluate the Limit

The limit \( \lim_{x \to \infty} \frac{1}{e^x} \) evaluates to 0, since \( e^x \to \infty \) exponentially faster than any linear or algebraic rate.

Conclusion of the Function Limit

Substitute the result back into the modified expression: \[ \lim_{x \to \infty} \left( \frac{x}{e^x} - 1 \right) = 0 - 1 = -1 \]Thus, the original limit \( \lim_{x \to \infty} \left( x - e^x \right) = -\infty \).

Key concepts

Indeterminate Forms

In calculus, indeterminate forms appear when evaluating limits that are not immediately clear. They manifest in several ways, such as \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( \infty - \infty \), and others. These forms suggest that the limits cannot be directly computed with straightforward substitution, requiring special techniques. For example, the limit in the given exercise \( \lim _{x \rightarrow \infty}(x-e^{x}) \) results in an \( \infty - \infty \) form. Such forms demand careful manipulation, often involving L'Hôpital's Rule, to find the actual value. Understand that identifying an indeterminate form is the first crucial step in solving limit problems.

Derivatives

Derivatives are a fundamental concept in calculus. They measure the rate at which a function changes as its input changes, often interpreted as the slope of the tangent line at any given point on a graph. In the context of L'Hôpital's Rule, calculating derivatives of the numerator and denominator is key. For example, in a limit problem like \( \frac{x}{e^x} \), the derivative of \( x \) is 1, as it is the rate of change for a linear function. Conversely, the derivative of \( e^x \) is \( e^x \) itself, due to the unique property of exponential functions.Altogether, the derivative process transforms the original problem, making it more straightforward to compute the limit.

Limits

Limits are used to understand the behavior of functions as they approach a certain point or infinity. This concept helps describe how a function behaves near boundaries or specific points, often indicating values that are not explicitly defined.When evaluating a limit like \( \lim_{x \to \infty}(x - e^x) \), direct substitution leads to an indeterminate form. Instead, transforming the expression can reveal a manageable form to evaluate. With the given exercise, rewriting the function as \( \frac{x}{e^x} - 1 \) allows for the application of L'Hôpital's Rule, streamlining the computation.Ultimately, understanding limits involves recognizing when direct evaluation is insufficient and when alternative methods—or rules—like L'Hôpital's might simplify the process.

Exponential Functions

Exponential functions are mathematical functions of the form \( e^x \), where \( e \) is a constant approximately equal to 2.71828. These functions grow incredibly quickly compared to linear functions.In the exercise at hand, the term \( e^x \) plays a significant role. As \( x \) approaches infinity, \( e^x \) increases at an exponential rate, overshadowing linear terms like \( x \). This is why, when rewriting \( x - e^x \) as \( \frac{x}{e^x} \), the exponential function dominates, simplifying the process of evaluating the limit.Recognizing the behavior of exponential functions helps in solving calculus problems, as their rapid growth often predicts outcomes in limit problems, influencing conclusions about convergence or divergence.