evaluate the limit. if the limit does not exist, enter dne. lim(x→ - 4) (x + 4)/(x³ + 64)

Explanation:

Step1: Factor the denominator

We know that (a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})). Here (a=x) and (b = 4), so (x^{3}+64=(x + 4)(x^{2}-4x + 16)). Then the limit (\lim_{x\rightarrow - 4}\frac{x + 4}{x^{3}+64}=\lim_{x\rightarrow - 4}\frac{x + 4}{(x + 4)(x^{2}-4x + 16)}).

Step2: Simplify the function

Cancel out the common factor ((x + 4)) (since (x\neq - 4) when taking the limit), we get (\lim_{x\rightarrow - 4}\frac{1}{x^{2}-4x + 16}).

Step3: Substitute (x=-4)

Substitute (x=-4) into (\frac{1}{x^{2}-4x + 16}), we have (\frac{1}{(-4)^{2}-4\times(-4)+16}=\frac{1}{16 + 16+16}=\frac{1}{48}).

Answer:

(\frac{1}{48})