Question

lim(x→2) (x³ + x² - 4x - 4)/(x⁴ - 16)
*hint: this is how you start setting it up.
lim(x→2) (x²(x + 1)-4(x + 1))/(x⁴ - 16)
*hint: after this you can do the difference of perfect squares in the bottom x⁴ can be separated into x²

Explanation:

Step1: Factor the expressions

We know that \(x^{4}-16=(x^{2} + 4)(x^{2}-4)=(x^{2}+4)(x + 2)(x - 2)\) and \(x^{2}(x + 1)-4(x + 1)=(x + 1)(x^{2}-4)=(x + 1)(x + 2)(x - 2)\). So the limit \(\lim_{x
ightarrow2}\frac{x^{2}(x + 1)-4(x + 1)}{x^{4}-16}=\lim_{x
ightarrow2}\frac{(x + 1)(x + 2)(x - 2)}{(x^{2}+4)(x + 2)(x - 2)}\)

Step2: Cancel out common factors

Cancel out the common factors \((x + 2)\) and \((x - 2)\) (since \(x
eq2\) when taking the limit), we get \(\lim_{x
ightarrow2}\frac{x + 1}{x^{2}+4}\)

Step3: Substitute \(x = 2\)

Substitute \(x=2\) into \(\frac{x + 1}{x^{2}+4}\), we have \(\frac{2+1}{2^{2}+4}=\frac{3}{4 + 4}=\frac{3}{8}\)

Answer:

\(\frac{3}{8}\)