https://file.aiquickdraw.com/mathbot/file/1725712656361vpo5pz7c.jpg

Use the graph of f shown to the right to find the limit. If​ necessary, state that the limit does not exist.

ModifyingBelow lim With x right arrow 4

​f(x)

To analyze the graph and find the limit of \( \lim_{x \to 4} f(x) \), let's examine the behavior of the function \( f(x) \) as \( x \) approaches 4 from both the left and the right sides.

From the image, here's what we can observe:

- As \( x \) approaches 4 from the left (i.e., \( x \to 4^- \)), the function \( f(x) \) seems to decrease without bound, which means it tends to \( -\infty \).
- As \( x \) approaches 4 from the right (i.e., \( x \to 4^+ \)), the function \( f(x) \) increases without bound, which means it tends to \( +\infty \).

Since the left-hand limit \( \lim_{x \to 4^-} f(x) = -\infty \) and the right-hand limit \( \lim_{x \to 4^+} f(x) = +\infty \) are not equal, we can conclude that:

\[
\lim_{x \to 4} f(x) \text{ does not exist}.
\]

Would you like more details on why this occurs? Here are some questions that can deepen your understanding:

1. What conditions must be met for a limit to exist at a specific point?
2. How do vertical asymptotes affect the behavior of limits?
3. What is the difference between one-sided and two-sided limits?
4. How does the concept of infinity play into the existence of limits?
5. Can a limit approach infinity and still be defined?

**Tip:** When examining graphs for limits, always check the behavior from both the left and the right sides of the point in question.

Use the graph of f shown to the right to find the limit. If necessary, state that the limit does not exist. ModifyingBelow lim With x right arrow 4 f(x)

Analyze the graph of f(x) to find the limit as x approaches 4. Understand the behavior of the function with vertical asymptotes and how one-sided limits contribute to determining if the limit exists. Suitable for high school students learning limits and calculus concepts.

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