To evaluate the limit

$\lim_{x \to 0} \frac{e^{3/x}}{x^2}$

using L'Hopital's Rule, we first check if this limit is in an indeterminate form of $\frac{\infty}{\infty}$ or $\frac{0}{0}$.

Step 1: Determine the Indeterminate Form

As $x \to 0$:

• $e^{3/x} \to \infty$ because $3/x \to \infty$ (when $x > 0$).
• $x^2 \to 0$.

Thus, the expression $\frac{e^{3/x}}{x^2}$ is of the form $\frac{\infty}{0}$, which is not directly an indeterminate form suitable for L'Hopital's rule.

However, if we approach the limit from the positive side, we still encounter $\infty/0$, which is not a form that allows L'Hopital's rule directly. The limit diverges to infinity:

$\lim_{x \to 0^+} \frac{e^{3/x}}{x^2} = \infty.$

Hence, the limit diverges. No application of L'Hopital's rule is possible since it is not in an indeterminate form $0/0$ or $\infty/\infty$.

Would you like a further breakdown or related examples? Here are 5 potential next questions:

1. What are some examples where L'Hopital's rule applies?
2. How does L'Hopital's rule work for forms other than $\frac{\infty}{\infty}$?
3. Can you provide examples where limits do not diverge?
4. How can we calculate limits using different methods, like Taylor series?
5. How do exponential functions behave near zero?

Tip: When applying L'Hopital's rule, always check the form first!