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.