The problem asks to evaluate the following limit:

$\lim_{x \to \infty} \frac{x^{14}}{e^x}$

We can attempt to solve this problem using L'Hôpital's Rule since both the numerator and denominator approach infinity as $x$ approaches infinity. L'Hôpital's Rule states that if we encounter an indeterminate form $\frac{\infty}{\infty}$, we can differentiate the numerator and denominator and then take the limit of the resulting expression.

Step 1: Check the form

The given expression is of the indeterminate form $\frac{\infty}{\infty}$ as $x \to \infty$, so we can apply L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

We will differentiate the numerator and denominator:

• Derivative of the numerator $x^{14}$ is $14x^{13}$.
• Derivative of the denominator $e^x$ is $e^x$.

Thus, the limit becomes:

$\lim_{x \to \infty} \frac{14x^{13}}{e^x}$

Step 3: Repeat L'Hôpital's Rule

We are still at the indeterminate form $\frac{\infty}{\infty}$, so we apply L'Hôpital's Rule again:

• Derivative of the numerator $14x^{13}$ is $182x^{12}$.
• Derivative of the denominator $e^x$ is still $e^x$.

This results in:

$\lim_{x \to \infty} \frac{182x^{12}}{e^x}$

Step 4: Continue differentiating

If we continue applying L'Hôpital's Rule repeatedly, each time we reduce the power of $x$ in the numerator by 1, while the denominator remains $e^x$. After applying L'Hôpital's Rule 14 times, we will end up with:

$\lim_{x \to \infty} \frac{14!}{e^x}$

Since $14!$ is a constant and $e^x$ grows exponentially as $x \to \infty$, the fraction approaches 0. Therefore:

$\lim_{x \to \infty} \frac{x^{14}}{e^x} = 0$

Final Answer:

The value of the limit is $\boxed{0}$.

Would you like more details about the solution or any related concepts? Here are some follow-up questions:

1. How does the exponential function grow compared to polynomial functions as $x$ increases?
2. What is the intuition behind why L'Hôpital's Rule works for indeterminate forms like $\frac{\infty}{\infty}$?
3. What other types of indeterminate forms can L'Hôpital's Rule be applied to?
4. How can we analyze limits of rational functions without L'Hôpital's Rule?
5. What happens to limits of other types of functions as $x$ approaches infinity?

Tip: Remember that exponential functions like $e^x$ grow much faster than any polynomial, so in limits involving both, the polynomial will usually be dominated by the exponential term.