https://file.aiquickdraw.com/mathbot/file/1725335110560ruhdww5a.jpg

The expression in the image is:

\[
\lim_{x \to 0} \frac{3x^2}{1 - e^x}
\]

To solve this limit, we can start by evaluating the limit directly:

1. **Substitute \( x = 0 \) into the expression**:

   \[
   \frac{3(0)^2}{1 - e^0} = \frac{0}{1 - 1} = \frac{0}{0}
   \]

   The result is an indeterminate form \(\frac{0}{0}\), so we need to apply L'Hôpital's rule.

2. **Apply L'Hôpital's rule**:

   L'Hôpital's rule states that if the limit results in an indeterminate form like \(\frac{0}{0}\), we can differentiate the numerator and the denominator separately and then take the limit:

   - Differentiate the numerator \( 3x^2 \) with respect to \( x \):
     \[
     \frac{d}{dx} (3x^2) = 6x
     \]

   - Differentiate the denominator \( 1 - e^x \) with respect to \( x \):
     \[
     \frac{d}{dx} (1 - e^x) = -e^x
     \]

   Now, the limit becomes:
   \[
   \lim_{x \to 0} \frac{6x}{-e^x}
   \]

3. **Evaluate the new limit**:

   Substitute \( x = 0 \) into the new expression:
   \[
   \frac{6(0)}{-e^0} = \frac{0}{-1} = 0
   \]

So, the limit is:

\[
\lim_{x \to 0} \frac{3x^2}{1 - e^x} = 0
\]

Would you like any further details or explanations? Here are some additional questions to consider:

1. How is L'Hôpital's rule applied when the limit results in \(\frac{\infty}{\infty}\)?
2. What are some alternative methods to solve limits without using L'Hôpital's rule?
3. How do you differentiate \( e^x \) and why is it the same as \( e^x \)?
4. What happens to the limit if the numerator had a higher power of \( x \)?
5. How can you verify the result using a series expansion for \( e^x \)?

**Tip**: L'Hôpital's rule is useful for handling limits that result in indeterminate forms, but it’s important to remember that the rule only applies when you have these specific forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

Learn how to solve the limit \( \lim_{x \to 0} \frac{3x^2}{1 - e^x} \) using symbolic notation. This problem involves applying L'Hôpital's rule to resolve an indeterminate form. Suitable for advanced high school students.

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