lim(x→2) (x³ + x² - 4x - 4)/(x⁴ - 16)\n*hint: this is how you start setting it up.\nlim(x→2) (x²(x + 1)-4(x…

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

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

Answer

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\rightarrow2}\frac{x^{2}(x + 1)-4(x + 1)}{x^{4}-16}=\lim_{x\rightarrow2}\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\neq2) when taking the limit), we get (\lim_{x\rightarrow2}\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})