Explain how Sind the value of each lrnit e) \( \lim _{x \rightarrow 2^{+}} \frac{(x+5)^{3}(x-1)^{2}}{(x+2)^{3}(x-2)^{2}} \)

We are given the limit   limₓ → 2⁺ [(x + 5)³ (x – 1)²] / [(x + 2)³ (x – 2)²]. Step 1. Substitute x = 2 into factors that are continuous at 2:  • (x + 5) becomes 2 + 5 = 7.  • (x – 1) becomes 2 – 1 = 1.  • (x + 2) becomes 2 + 2 = 4. So, if we ignore the (x – 2) factor, the numerator goes to 7³ · 1² = 343 and the (x + 2)³ factor goes to 4³ = 64. Step 2. Examine the problematic factor (x – 2)²:  Since x → 2⁺ (approaching 2 from the right), x exceeds 2 by a small positive amount. Therefore, (x – 2) is a small positive number. Squaring it, (x – 2)², gives a small number that trends to 0 as x approaches 2 from the right. Step 3. Combine the observations:  The numerator tends to the finite number 343 (multiplied by additional nonzero constants we found), and the denominator tends to 64 multiplied by (x – 2)², which goes to 64·0 = 0⁺. Step 4. Determine the limit:  As the denominator shrinks to 0⁺ while the numerator remains a positive constant, the fraction grows without bound. Thus, the limit is +∞. Final Answer: → +∞