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.