L'Hospital's Rule: Evaluating A Limit Problem

In calculus, evaluating limits is a fundamental concept. Often, we encounter situations where direct substitution leads to indeterminate forms like 0/0 or ∞/∞. In such cases, L'Hôpital's Rule becomes an invaluable tool. Let's explore how to use L'Hôpital's Rule to evaluate the limit: $\lim _{x \rightarrow \infty} 13 x (e^{\frac{1}{x}}-1)$. This exploration will provide a clear, step-by-step approach to solving this type of problem, ensuring a solid understanding of both the theoretical underpinnings and practical applications.

Understanding the Indeterminate Form

The initial step in tackling this limit involves recognizing why direct substitution fails. As x approaches infinity, the term 1/x approaches zero. Consequently, e^1/x approaches e⁰ which equals 1. Therefore, the expression inside the parenthesis (e^1/x - 1) approaches 0. At the same time, the term 13x approaches infinity. This leads to an indeterminate form of the type ∞ * 0. Indeterminate forms signal that we cannot determine the limit's value through simple substitution and that further analysis, possibly using L'Hôpital's Rule, is required. Recognizing this indeterminate form is crucial because it justifies the application of more advanced techniques to find the limit. This initial assessment sets the stage for a more rigorous mathematical approach.

Rewriting the Expression

The key to applying L'Hôpital's Rule to the limit $\lim _{x \rightarrow \infty} 13 x (e^{\frac{1}{x}}-1)$ lies in transforming the expression into a suitable form, specifically a fraction that results in either 0/0 or ∞/∞ when x approaches infinity. Currently, the expression is in the form of a product, which is not directly amenable to L'Hôpital's Rule. To convert it into a fraction, we can rewrite the expression as follows:

$\lim _{x \rightarrow \infty} 13 x (e^{\frac{1}{x}}-1) = \lim _{x \rightarrow \infty} \frac{e^{\frac{1}{x}}-1}{\frac{1}{13x}}$

By doing this, we transform the original problem into a fraction. As x approaches infinity, the numerator (e^1/x - 1) approaches 0, and the denominator (1/13x) also approaches 0. This yields an indeterminate form of 0/0, which is perfect for applying L'Hôpital's Rule. This algebraic manipulation is a critical step, as it sets the stage for the subsequent differentiation and evaluation. Recognizing and executing this transformation correctly is vital for the successful application of L'Hôpital's Rule.

Applying L'Hôpital's Rule

Now that we have the limit in the form 0/0, we can apply L'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value is in an indeterminate form (0/0 or ∞/∞), then the limit is equal to the limit of the derivatives of f(x) and g(x), i.e., $\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists. In our case, we need to find the derivatives of the numerator and the denominator of our rewritten expression. This involves differentiating e^1/x - 1 and 1/13x with respect to x. The derivative of e^1/x - 1 requires using the chain rule, while the derivative of 1/13x is a straightforward application of the power rule. Accurate differentiation is paramount, as any errors here will propagate through the rest of the solution. Therefore, careful attention to detail and a solid understanding of differentiation techniques are essential for correctly applying L'Hôpital's Rule.

Calculating the Derivatives

To successfully apply L'Hôpital's Rule, we need to compute the derivatives of both the numerator and the denominator of our expression. Let's start with the numerator, f(x) = e^1/x - 1. To find f'(x), we'll use the chain rule. The chain rule states that if we have a composite function, the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. In this case, the outer function is e^u, and the inner function is u = 1/x. The derivative of e^u with respect to u is simply e^u, and the derivative of 1/x with respect to x is -1/x². Therefore, applying the chain rule gives us:

f'(x) = e^1/x * (-1/x²) = -e^1/x/x²

Now, let's find the derivative of the denominator, g(x) = 1/13x. This can be rewritten as g(x) = (1/13)x^-1. Using the power rule, which states that the derivative of xⁿ is nx^n-1, we get:

g'(x) = (1/13) * (-1) * x^-2 = -1/(13x²)

These derivatives are crucial for the next step in applying L'Hôpital's Rule. Ensuring their accuracy is paramount, as they form the foundation for the subsequent limit evaluation.

Evaluating the Limit of the Derivatives

After finding the derivatives, the next step is to evaluate the limit of the ratio of these derivatives as x approaches infinity. We have:

$\lim _{x \rightarrow \infty} \frac{f'(x)}{g'(x)} = \lim _{x \rightarrow \infty} \frac{-e^{\frac{1}{x}}/x^2}{-1/(13x^2)}$

To simplify this expression, we can multiply the numerator and denominator by -13x². This simplifies the fraction to:

$\lim _{x \rightarrow \infty} 13e^{\frac{1}{x}}$

Now, as x approaches infinity, 1/x approaches 0. Therefore, e^1/x approaches e⁰, which equals 1. Thus, the limit becomes:

$\lim _{x \rightarrow \infty} 13e^{\frac{1}{x}} = 13 * 1 = 13$

This result indicates that the limit of the original expression as x approaches infinity is 13. The simplification and subsequent evaluation are critical steps, demonstrating the power and elegance of L'Hôpital's Rule in resolving indeterminate forms.

Conclusion

In conclusion, by recognizing the indeterminate form, rewriting the expression, applying L'Hôpital's Rule, and carefully evaluating the derivatives, we've successfully determined that $\lim _{x \rightarrow \infty} 13 x (e^{\frac{1}{x}}-1) = 13$. This exercise highlights the importance of L'Hôpital's Rule as a powerful technique in calculus for evaluating limits that initially appear unsolvable through direct substitution. Understanding and mastering this rule is essential for anyone studying calculus, as it provides a systematic approach to handling indeterminate forms and finding limits with precision. Remember to always verify the indeterminate form before applying the rule and to double-check your derivatives to ensure accuracy.

For further learning and a deeper understanding of L'Hôpital's Rule, you can explore resources like Khan Academy's L'Hôpital's Rule section.