Using L'Hopital rule, evaluate limx→0 e^3/x divided by x^2

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!

Learn how to evaluate the limit of (e^3/x) / x^2 as x approaches 0 using L'Hopital's Rule. This problem involves the concepts of limits, exponential functions, and indeterminate forms, and is ideal for students in advanced calculus courses.

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