Evaluating Limits With L'Hôpital's Rule Lim X→∞ 15x^2/e^{4x}

Introduction

In the realm of calculus, evaluating limits stands as a fundamental concept, crucial for understanding the behavior of functions as their input approaches a specific value or infinity. Among the diverse techniques available for limit evaluation, L'Hôpital's Rule emerges as a powerful tool, particularly adept at handling indeterminate forms. This article delves into the application of L'Hôpital's Rule to evaluate the limit ${\lim_{x \rightarrow \infty} \frac{15x^2}{e^{4x}}}$, providing a comprehensive understanding of the process and the underlying principles.

This exploration will not only demonstrate the step-by-step application of L'Hôpital's Rule but also shed light on the conditions under which it can be applied effectively. We'll dissect the indeterminate form, verify the prerequisites for L'Hôpital's Rule, and then systematically apply the rule until we arrive at a determinate form, allowing us to confidently evaluate the limit. By the end of this discourse, you'll have a solid grasp of how to tackle similar limit problems with confidence and precision. This detailed approach is essential for mastering calculus and its applications in various fields.

Understanding Indeterminate Forms and L'Hôpital's Rule

Before diving into the specific problem, it's crucial to grasp the concept of indeterminate forms. Indeterminate forms arise when directly substituting the limit value into a function results in expressions like ${\frac{0}{0}}$, ${\frac{\infty}{\infty}}$, ${0 \cdot \infty}$, ${\infty - \infty}$, ${0^0}$, ${1^\infty}$, or ${\infty^0}$. These forms don't immediately reveal the limit's value, necessitating techniques like L'Hôpital's Rule to resolve them.

L'Hôpital's Rule provides a method for evaluating limits of indeterminate forms ${\frac{0}{0}}$ or ${\frac{\infty}{\infty}}$. The rule states that if ${\lim_{x \rightarrow c} f(x) = 0}$ and ${\lim_{x \rightarrow c} g(x) = 0}$ (or both limits are ${\pm \infty}$), and if ${\lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}}$ exists, then ${\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}}$. Here, ${f'(x)}$ and ${g'(x)}$ represent the derivatives of ${f(x)}$ and ${g(x)}$, respectively. The key idea is that by differentiating the numerator and the denominator, we can sometimes simplify the expression and reveal the limit's true value. It is imperative to verify that the conditions for L'Hôpital's Rule are met before applying it, ensuring the validity of the result. This rule transforms the problem into a potentially simpler limit, making it a cornerstone technique in calculus. Mastering this rule is critical for anyone delving into advanced mathematical concepts and applications.

Problem Statement: ${\lim_{x \rightarrow \infty} \frac{15x^2}{e^{4x}}}$

Now, let's focus on the specific limit we aim to evaluate:

$$\lim_{x \rightarrow \infty} \frac{15x^2}{e^{4x}}$$

Our objective is to determine the behavior of the function ${\frac{15x^2}{e^{4x}}}$ as x approaches infinity. A direct substitution of infinity into the function yields an indeterminate form of ${\frac{\infty}{\infty}}$, which signals the potential applicability of L'Hôpital's Rule. Before we proceed, it's crucial to formally verify that this is indeed an indeterminate form. As x grows without bound, both the numerator, 15x², and the denominator, e^(4x), also grow without bound. This confirms that we are dealing with the indeterminate form ${\frac{\infty}{\infty}}$, making L'Hôpital's Rule a viable strategy. The presence of this indeterminate form is the key indicator that we need a more sophisticated method to find the limit. Recognizing this form early on is essential for efficient problem-solving in calculus. With the indeterminate form identified, we can now confidently proceed to apply L'Hôpital's Rule to this limit.

Step-by-Step Application of L'Hôpital's Rule

Step 1: First Application of the Rule

To apply L'Hôpital's Rule, we first need to compute the derivatives of the numerator and the denominator of our function, (\frac{15x^2}{e{4x}}. The derivative of the numerator, 15x², with respect to x is 30x. The derivative of the denominator, e^(4x), with respect to x is 4e^(4x). Thus, we have:

$$\frac{d}{dx}(15x^2) = 30x$$

$$\frac{d}{dx}(e^{4x}) = 4e^{4x}$$

Applying L'Hôpital's Rule, we replace the original limit with the limit of the ratio of these derivatives:

$$\lim_{x \rightarrow \infty} \frac{15x^2}{e^{4x}} = \lim_{x \rightarrow \infty} \frac{30x}{4e^{4x}}$$

However, if we try to evaluate this new limit by direct substitution, we still encounter an indeterminate form of ${\frac{\infty}{\infty}}$. This means we need to apply L'Hôpital's Rule again to further simplify the expression. This iterative process is a common characteristic of problems requiring L'Hôpital's Rule, particularly when dealing with exponential functions and polynomials. The first application of the rule has simplified the problem, but further application is necessary to reach a determinate form.

Step 2: Second Application of the Rule

Since the limit ${\lim_{x \rightarrow \infty} \frac{30x}{4e^{4x}}}$ still results in the indeterminate form ${\frac{\infty}{\infty}}$, we apply L'Hôpital's Rule a second time. This involves finding the derivatives of the new numerator, 30x, and the new denominator, 4e^(4x). The derivative of 30x with respect to x is simply 30. The derivative of 4e^(4x) with respect to x is 16e^(4x). Therefore, we have:

$$\frac{d}{dx}(30x) = 30$$

$$\frac{d}{dx}(4e^{4x}) = 16e^{4x}$$

Applying L'Hôpital's Rule again, we get:

$$\lim_{x \rightarrow \infty} \frac{30x}{4e^{4x}} = \lim_{x \rightarrow \infty} \frac{30}{16e^{4x}}$$

Now, let's examine the limit ${\lim_{x \rightarrow \infty} \frac{30}{16e^{4x}}}$. As x approaches infinity, the denominator 16e^(4x) also approaches infinity, while the numerator remains constant at 30. This situation no longer represents an indeterminate form, and we can directly evaluate the limit.

Evaluating the Resulting Limit

After the second application of L'Hôpital's Rule, we arrived at the limit:

$$\lim_{x \rightarrow \infty} \frac{30}{16e^{4x}}$$

To evaluate this limit, we analyze the behavior of the numerator and denominator as x approaches infinity. The numerator is a constant value, 30. The denominator, 16e^(4x), grows without bound as x approaches infinity because the exponential function e^(4x) increases rapidly. Therefore, we have a constant divided by a quantity that is approaching infinity. In such cases, the limit approaches zero.

Mathematically, this can be expressed as:

$$\lim_{x \rightarrow \infty} \frac{30}{16e^{4x}} = 0$$

This result indicates that as x becomes infinitely large, the value of the fraction ${\frac{30}{16e^{4x}}}$ gets infinitesimally small, approaching zero. This is a crucial step in the overall evaluation, as it transforms the indeterminate form into a determinate one, allowing us to conclude the limit's value. This outcome underscores the power of L'Hôpital's Rule in simplifying complex limits to a point where they can be directly evaluated.

Conclusion: The Limit Evaluated

By systematically applying L'Hôpital's Rule twice, we have successfully evaluated the limit ${\lim_{x \rightarrow \infty} \frac{15x^2}{e^{4x}}}$. We started by identifying the indeterminate form ${\frac{\infty}{\infty}}$, which justified the use of L'Hôpital's Rule. Each application of the rule involved taking the derivatives of the numerator and the denominator, simplifying the expression until we reached a determinate form.

Through this process, we found that:

$$\lim_{x \rightarrow \infty} \frac{15x^2}{e^{4x}} = 0$$

This result demonstrates that as x approaches infinity, the exponential function e^(4x) grows much faster than the polynomial function 15x². Consequently, the ratio ${\frac{15x^2}{e^{4x}}}$ approaches zero. This problem serves as a classic example of how L'Hôpital's Rule can be effectively used to evaluate limits involving indeterminate forms, particularly those involving exponential and polynomial functions. The detailed step-by-step approach highlights the importance of careful application and verification of the rule's conditions for accurate results. Mastering this technique is crucial for students and professionals in mathematics, engineering, and related fields.