Dada la función 𝑓(𝑥) = {𝑒𝑥2−9    𝑠𝑖 𝑥 < 3
 𝑥2−4𝑥
 𝑥2−3

It seems that the function provided is incomplete. However, based on the context and the previous content, I will reconstruct and analyze the problem to the best of my ability.

The piecewise function \( f(x) \) appears to be defined as follows:

\[
f(x) = 
\begin{cases} 
e^{x^2 - 9} & \text{if } x < 3 \\
\frac{x^2 - 4x}{x^2 - 3x - 4} & \text{if } x \geq 3
\end{cases}
\]

With this function, we are asked to calculate the following limits:

1. \( \lim_{x \to 3} f(x) \)
2. \( \lim_{x \to 0} f(x) \)

### Limit as \( x \) approaches 3:
For \( x \to 3 \), the behavior of the function depends on whether \( x \) approaches from the left (\( x < 3 \)) or from the right (\( x \geq 3 \)).

- **For \( x \) approaching 3 from the left**:
  \( f(x) = e^{x^2 - 9} \)
  
  Substitute \( x = 3 \):
  \[
  \lim_{x \to 3^-} f(x) = e^{3^2 - 9} = e^{9 - 9} = e^0 = 1
  \]

- **For \( x \) approaching 3 from the right**:
  \( f(x) = \frac{x^2 - 4x}{x^2 - 3x - 4} \)
  
  Factor the numerator and the denominator:
  \[
  x^2 - 4x = x(x - 4)
  \]
  \[
  x^2 - 3x - 4 = (x - 4)(x + 1)
  \]
  Therefore:
  \[
  f(x) = \frac{x(x - 4)}{(x - 4)(x + 1)} = \frac{x}{x + 1}, \quad \text{for } x \neq 4
  \]

  Substitute \( x = 3 \):
  \[
  \lim_{x \to 3^+} f(x) = \frac{3}{3 + 1} = \frac{3}{4}
  \]

Since the left-hand limit (\( 1 \)) and the right-hand limit (\( \frac{3}{4} \)) as \( x \) approaches 3 are not equal, the limit does not exist at \( x = 3 \).

### Limit as \( x \) approaches 0:
For \( x \to 0 \), we use the first case of the piecewise function because \( 0 < 3 \).

\[
\lim_{x \to 0} f(x) = \lim_{x \to 0} e^{x^2 - 9} = e^{0^2 - 9} = e^{-9}
\]

Thus, \( \lim_{x \to 0} f(x) = e^{-9} \).

Would you like a more detailed explanation or have any other questions?

Here are five related questions to explore further:

1. What conditions must be met for a limit to exist at a point?
2. How can we verify if a piecewise function is continuous at a given point?
3. What are the implications of a non-existent limit on the continuity of a function?
4. How does factoring help in simplifying limit problems?
5. What is the significance of the exponential function in limit calculations?

**Tip:** Always check both left-hand and right-hand limits when dealing with piecewise functions to determine if the overall limit exists.

Explore the analysis of limits for the piecewise function f(x) = e^(x^2 - 9) if x < 3, (x^2 - 4x) / (x^2 - 3x - 4) if x >= 3. Understand how to calculate limits approaching specific points and the implications of non-existent limits. Suitable for advanced high school students and those interested in calculus and piecewise functions.

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