\displaystyle\lim_{x\to\infty}\dfrac{e^{x^2}}{x^4}=?

To find the limit \(\displaystyle\lim_{x\to\infty}\dfrac{e^{x^2}}{x^4}\), we can analyze the behavior of the numerator and denominator as \( x \) approaches infinity.

### Step 1: Analyze the Growth Rates
- The numerator \( e^{x^2} \) grows extremely fast because exponential functions generally grow much faster than polynomial functions.
- The denominator \( x^4 \) grows at a polynomial rate, which is much slower than the exponential growth of \( e^{x^2} \).

### Step 2: Determine the Limit
Given that the numerator grows exponentially faster than the polynomial denominator, the ratio will tend to infinity as \( x \) approaches infinity.

Thus, we can conclude that:
\[
\lim_{x\to\infty}\dfrac{e^{x^2}}{x^4} = \infty
\]

The limit is \(\infty\).

Would you like further explanation or additional details?

Here are five related questions:
1. What is the limit \(\lim_{x \to \infty} \frac{e^x}{x^n}\) for any positive integer \(n\)?
2. How would the limit change if the denominator were \(x^2\) instead of \(x^4\)?
3. Can you apply L'Hôpital's Rule to evaluate the limit, and what does it show?
4. What is the limit \(\lim_{x \to \infty} \frac{x^4}{e^{x^2}}\)?
5. How would you compare the growth rates of \(e^{x^2}\) and \(x^n\) for any polynomial \(n\)?

**Tip**: Exponential functions like \(e^{x^2}\) grow much faster than any polynomial as \(x\) tends to infinity.

Explore how to evaluate the limit of e^(x^2) / x^4 as x approaches infinity. Understand the growth rates of exponential and polynomial functions and how they affect the limit. Suitable for college-level mathematics enthusiasts.

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