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

To evaluate   limₓ→₋₂⁻ [(x + 5)³ (x – 1)²]⁄[(x + 2)³ (x – 2)²], follow these steps: 1. Recognize that as x → –2, most factors in the expression approach finite, nonzero numbers except for the factor (x + 2) in the denominator. Since (x + 2) is raised to the power 3, and x → –2 (from the left), this factor drives the behavior of the limit. 2. Determine the value of each factor when x = –2:   • (x + 5)³ becomes (–2 + 5)³ = 3³ = 27.   • (x – 1)² becomes (–2 – 1)² = (–3)² = 9.   • (x – 2)² becomes (–2 – 2)² = (–4)² = 16.   Thus, the factors in the numerator and the (x – 2)² factor in the denominator are nonzero and contribute the finite value 27 × 9 and 16, respectively. 3. Analyze the problematic factor (x + 2)³:   • As x → –2 from the left, denote a small positive number h = –2 – x (so that h > 0 and when x → –2⁻, h → 0⁺). Then, x + 2 = –h, which is a small negative number.   • Raised to the power 3, (x + 2)³ becomes (–h)³ = –h³.   • Since h is very small and positive, h³ is also very small and positive; its negative, –h³, approaches 0⁻ (zero from the negative side). 4. Rewrite the limit in terms of these evaluations:   The numerator tends to 27 × 9 = 243, and the factor (x – 2)² tends to 16, so the finite part overall is 243⁄16.   Thus, the overall expression is approximately     (243/16)⁄[(x + 2)³] ≈ (243/16)⁄(–h³)   as x → –2⁻. 5. Conclude the behavior of the limit:   Since (–h³) is a small negative number, dividing a positive constant by a number that approaches 0 through negative values results in a limit of –∞. Therefore, the limit is –∞.