5. evaluate this limit.\nlim(x→0) ((4 + x)^2 - 16)/x

Answer

Explanation:

Step1: Expand the numerator

Expand ((4 + x)^{2}) using ((a + b)^{2}=a^{2}+2ab + b^{2}), so ((4 + x)^{2}=16+8x+x^{2}). Then the expression becomes (\lim_{x\rightarrow0}\frac{16 + 8x+x^{2}-16}{x}).

Step2: Simplify the numerator

(16 + 8x+x^{2}-16=8x+x^{2}), so the limit is (\lim_{x\rightarrow0}\frac{8x+x^{2}}{x}).

Step3: Factor out x from the numerator

(\frac{8x+x^{2}}{x}=\frac{x(8 + x)}{x}), and for (x\neq0) we can cancel out the (x) terms, getting (\lim_{x\rightarrow0}(8 + x)).

Step4: Evaluate the limit

Substitute (x = 0) into (8 + x), we get (8+0 = 8).