Evaluating $\lim _{x \rightarrow \infty} \frac{8 X^2}{e^{10 X}}$ Using L'Hopital's Rule

In the realm of calculus, evaluating limits is a fundamental concept, particularly when dealing with indeterminate forms. Among the powerful tools available, L'Hôpital's Rule stands out as a robust method for resolving limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. This article delves into the application of L'Hôpital's Rule to evaluate the limit $\lim _{x \rightarrow \infty} \frac{8 x^2}{e^{10 x}}$, providing a comprehensive understanding of the process and the underlying principles.

Understanding the Indeterminate Form

Before applying L'Hôpital's Rule, it's crucial to recognize the nature of the limit. As $x$ approaches infinity, both the numerator, $8x^2$, and the denominator, $e^{10x}$, tend towards infinity. This results in the indeterminate form $\frac{\infty}{\infty}$, which signifies that the limit cannot be determined directly by substitution. Indeterminate forms necessitate the use of techniques like L'Hôpital's Rule to unveil the true behavior of the function as $x$ approaches its limit. Recognizing these forms is the first step in correctly applying limit evaluation techniques. The presence of an indeterminate form signals the need for further analysis, often involving algebraic manipulation or the application of specific rules like L'Hôpital's. Correctly identifying the indeterminate form is crucial for selecting the appropriate method for limit evaluation.

L'Hôpital's Rule: A Powerful Tool

L'Hôpital's Rule provides a systematic approach to evaluating limits of indeterminate forms. It states that if $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$ (or if both limits are $\pm \infty$), and if $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ exists, then

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.$$

In essence, the rule allows us to differentiate the numerator and denominator separately and then re-evaluate the limit. This process can be repeated if the new limit is also an indeterminate form. The power of L'Hôpital's Rule lies in its ability to simplify complex limits by reducing them to simpler forms through differentiation. However, it's crucial to ensure that the conditions for applying the rule are met before proceeding. Incorrect application can lead to erroneous results. The rule is particularly useful when dealing with ratios of functions that become increasingly complex near the limit point. By differentiating, the complexity can often be reduced, making the limit evaluation more manageable.

Applying L'Hôpital's Rule to $\lim _{x \rightarrow \infty} \frac{8 x^2}{e^{10 x}}$

Let's apply L'Hôpital's Rule to our given limit, $\lim _{x \rightarrow \infty} \frac{8 x^2}{e^{10 x}}$.

1. Identify the functions: Let $f(x) = 8x^2$ and $g(x) = e^{10x}$.

2. Check for the indeterminate form: As $x \rightarrow \infty$, $f(x) \rightarrow \infty$ and $g(x) \rightarrow \infty$, confirming the indeterminate form $\frac{\infty}{\infty}$.

3. Differentiate:

   • $f'(x) = 16x$
   • $g'(x) = 10e^{10x}$

4. Apply L'Hôpital's Rule:

   $$\lim _{x \rightarrow \infty} \frac{8 x^2}{e^{10 x}} = \lim _{x \rightarrow \infty} \frac{16 x}{10 e^{10 x}}$$

The new limit, $\lim _{x \rightarrow \infty} \frac{16 x}{10 e^{10 x}}$, is still in the indeterminate form $\frac{\infty}{\infty}$. This necessitates the repeated application of L'Hôpital's Rule. The initial application of L'Hôpital's Rule has simplified the expression, but the indeterminate form persists, indicating the need for further differentiation. This iterative process is a common characteristic when dealing with certain types of limits. Each application of the rule aims to reduce the complexity of the expression until a determinate form is achieved.

Second Application of L'Hôpital's Rule

Since we still have an indeterminate form, we apply L'Hôpital's Rule again.

1. Differentiate again:

   • $f''(x) = 16$
   • $g''(x) = 100e^{10x}$

2. Apply L'Hôpital's Rule:

   $$\lim _{x \rightarrow \infty} \frac{16 x}{10 e^{10 x}} = \lim _{x \rightarrow \infty} \frac{16}{100 e^{10 x}}$$

Now, the limit becomes $\lim _{x \rightarrow \infty} \frac{16}{100 e^{10 x}}$. As $x$ approaches infinity, the denominator, $100e^{10x}$, approaches infinity, while the numerator remains constant at 16. This yields a determinate form, allowing us to directly evaluate the limit. The repeated application of L'Hôpital's Rule has transformed the limit into a form where the behavior as $x$ approaches infinity is readily apparent. This demonstrates the power of the rule in handling complex limits that initially appear intractable. The process of differentiation has effectively reduced the complexity, leading to a clear and concise resolution.

Evaluating the Final Limit

As $x$ approaches infinity, $e^{10x}$ approaches infinity. Therefore, the fraction $\frac{16}{100 e^{10 x}}$ approaches 0.

$$\lim _{x \rightarrow \infty} \frac{16}{100 e^{10 x}} = 0$$

Thus, we have successfully evaluated the limit using L'Hôpital's Rule.

Conclusion

By applying L'Hôpital's Rule twice, we have shown that $\lim _{x \rightarrow \infty} \frac{8 x^2}{e^{10 x}} = 0$. This example illustrates the effectiveness of L'Hôpital's Rule in handling indeterminate forms and evaluating limits that would otherwise be difficult to compute. The key to successful application lies in recognizing the indeterminate form, correctly differentiating the numerator and denominator, and repeating the process as needed until a determinate form is achieved. L'Hôpital's Rule is a cornerstone technique in calculus, providing a powerful tool for understanding the behavior of functions near their limits. Its application extends beyond simple examples, playing a crucial role in solving complex problems in various fields of science and engineering. Mastering L'Hôpital's Rule enhances one's ability to analyze and interpret the behavior of mathematical functions, contributing to a deeper understanding of calculus and its applications.

Key Takeaways

• L'Hôpital's Rule is applicable to limits in the indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$.
• The rule involves differentiating the numerator and denominator separately.
• The process can be repeated if the new limit is also an indeterminate form.
• Careful application and correct differentiation are crucial for obtaining accurate results.
• This method simplifies complex limit evaluations by reducing expressions to more manageable forms.
• Understanding L'Hôpital's Rule is essential for advanced calculus and related fields.

By understanding and applying L'Hôpital's Rule, we gain a valuable tool for navigating the intricacies of limit evaluation and unlocking a deeper understanding of calculus concepts.