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Let's analyze the given graph of the function \( f(x) \) and determine the limits specified in the question.

### a) \( \lim_{x \to \infty} f(x) \)
As \( x \to \infty \), we observe that the graph of \( f(x) \) increases without bound. Therefore:

\[
\lim_{x \to \infty} f(x) = \infty
\]

### b) \( \lim_{x \to -\infty} f(x) \)
As \( x \to -\infty \), the function appears to decrease indefinitely, moving downward. Thus:

\[
\lim_{x \to -\infty} f(x) = -\infty
\]

### c) \( \lim_{x \to -1^{-}} f(x) \) (the left-hand limit as \( x \to -1 \))
Approaching \( x = -1 \) from the left side, the function appears to be approaching the value \( y = 4 \). Hence:

\[
\lim_{x \to -1^{-}} f(x) = 4
\]

### d) \( \lim_{x \to -1^{+}} f(x) \) (the right-hand limit as \( x \to -1 \))
From the right side, as \( x \to -1 \), the graph also approaches \( y = 4 \). So:

\[
\lim_{x \to -1^{+}} f(x) = 4
\]

Since the left-hand and right-hand limits are equal, we can conclude:

\[
\lim_{x \to -1} f(x) = 4
\]

### e) \( \lim_{x \to 5} f(x) \)
At \( x = 5 \), the graph shows that the function is continuous and approaches the value \( y = 7 \). Therefore:

\[
\lim_{x \to 5} f(x) = 7
\]

### Summary of Limits:
- \( \lim_{x \to \infty} f(x) = \infty \)
- \( \lim_{x \to -\infty} f(x) = -\infty \)
- \( \lim_{x \to -1^{-}} f(x) = 4 \)
- \( \lim_{x \to -1^{+}} f(x) = 4 \)
- \( \lim_{x \to 5} f(x) = 7 \)

Let me know if you would like further clarifications or details!

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Here are some related questions to consider:
1. What is the definition of a two-sided limit?
2. How do we determine if a function has a discontinuity at a certain point?
3. What is the difference between one-sided and two-sided limits?
4. How would the limits change if there were an asymptote in the graph?
5. Can limits exist at points where the function is undefined?

**Tip:** When dealing with limits approaching infinity, the behavior of the graph (increasing or decreasing without bound) is crucial for determining the limit.

A partir de la gráfica f(x), determine los límites: lim_{x -> ∞} f(x), lim_{x -> -∞} f(x), lim_{x -> -1^-} f(x), lim_{x -> -1^+} f(x), lim_{x -> 5} f(x).

Analyze the graph of f(x) and determine limits, including left-hand and right-hand limits, and limits as x approaches infinity. Suitable for high school students studying calculus and graph analysis.

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