Explain how to find the value of each limit d) \( \lim _{x \rightarrow-2^{-}} \frac{(x+5)^{3}(x-1)^{2}}{(x+2)^{3}(x-2)^{2}} \)

To find the limit as \( x \) approaches \(-2\) from the left for the given expression, we need to analyze the behavior of both the numerator and the denominator. As \( x \) approaches \(-2\), the term \( (x+2)^3 \) in the denominator approaches zero and will dominate the denominator’s behavior, while the rest of the terms can be evaluated at \( x = -2 \). The numerator \( (x+5)^3(x-1)^2 \) does not approach zero as fast since both \( (x+5) \) and \( (x-1) \) are non-zero at \( x = -2\). Evaluating these components at \( x = -2 \): - The numerator becomes \( (3)^3(1)^2 = 27 \). - The denominator approaches \( 0 \) but is slightly negative due to \( \lim _{x \rightarrow -2^{-}} (x + 2)^3 \). So, as \( x \to -2^{-} \), the limit tends towards \(-\infty\) due to a positive numerator and a negative denominator. Thus, the limit is \[ \lim_{x \to -2^{-}} \frac{(x+5)^{3}(x-1)^{2}}{(x+2)^{3}(x-2)^{2}} = -\infty. \] Now, don’t you just love limits? They teach us to navigate the tricky waters where functions throw a curveball! They help us understand continuity and how shapes behave near critical points. Isn't math the ultimate puzzle? You can further your exploration into limits with classic calculus resources like "Thomas' Calculus" or "Calculus" by James Stewart. Each of these books delves into the inspiring world of analysis and offers insightful examples alongside a treasury of practice problems to sharpen those math skills. Happy calculating!