Question

1. evaluate this limit.

lim(x→0) ((4 + x)^2 - 16)/x

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 \(\lim_{x
ightarrow0}\frac{16 + 8x+x^{2}-16}{x}\).

Step2: Simplify the numerator

\(16 + 8x+x^{2}-16=8x+x^{2}\), so the limit is \(\lim_{x
ightarrow0}\frac{8x+x^{2}}{x}\).

Step3: Factor out x from the numerator

\(\frac{8x+x^{2}}{x}=\frac{x(8 + x)}{x}\), and for \(x
eq0\) we can cancel out the \(x\) terms, getting \(\lim_{x
ightarrow0}(8 + x)\).

Step4: Evaluate the limit

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

Answer:

8