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evaluate the given limits using the graph

To evaluate the given limits, we are working with the function:

\[
f(x) = \frac{1}{(x + 1)^2}
\]

and need to analyze the following limits:

- (a) \( \lim_{x \to -1^-} f(x) \)
- (b) \( \lim_{x \to -1^+} f(x) \)

### Explanation of the limits:

1. **(a) Left-hand limit \( \lim_{x \to -1^-} f(x) \)**:
   - This is the limit as \(x\) approaches \(-1\) from the left side.
   - As \( x \to -1^- \), \( (x + 1) \to 0^- \), meaning the term inside the parentheses approaches \(0\) from the negative side.
   - Since we are squaring \( (x+1)^2 \), this negative value becomes positive, and as \( (x + 1)^2 \to 0^+ \), \( f(x) \to \infty \).
   - Thus, \( \lim_{x \to -1^-} f(x) = \infty \).

2. **(b) Right-hand limit \( \lim_{x \to -1^+} f(x) \)**:
   - This is the limit as \(x\) approaches \(-1\) from the right side.
   - As \( x \to -1^+ \), \( (x + 1) \to 0^+ \), meaning the term inside the parentheses approaches \(0\) from the positive side.
   - Again, squaring this term results in a positive value, and as \( (x + 1)^2 \to 0^+ \), \( f(x) \to \infty \).
   - Thus, \( \lim_{x \to -1^+} f(x) = \infty \).

### Conclusion:
Both the left-hand and right-hand limits tend to infinity. Therefore:
- \( \lim_{x \to -1^-} f(x) = \infty \)
- \( \lim_{x \to -1^+} f(x) = \infty \)

Let me know if you'd like more details, and here are some related questions to expand your understanding:

1. What happens to the function as \(x\) approaches \( -1 \) in general (without specifying direction)?
2. How would the limits change if the function was \( \frac{1}{(x+1)^3} \)?
3. What does it mean graphically when a function approaches infinity as \(x\) approaches a certain value?
4. Can we determine whether the function has any vertical asymptotes from the given limits?
5. How do left-hand and right-hand limits relate to the concept of continuity?

**Tip**: When dealing with rational functions, it's often helpful to determine whether there's a vertical asymptote at certain points by checking where the denominator becomes zero.

Evaluate the limits for the function f(x) = 1 / (x + 1)^2 as x approaches -1 from both the left and right.

Understand how to evaluate the limits of the function f(x) = 1 / (x + 1)^2 as x approaches -1 from both the left and right. The step-by-step solution explains concepts of rational functions, limits, and asymptotes. Ideal for high school students in Grades 10-12.

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