attempt 1: 2 attempts remaining. let f(x) = (x³ - 64)/(x - 4). find lim f(x) as x→4. enter i for ∞, -i for…
attempt 1: 2 attempts remaining. let f(x) = (x³ - 64)/(x - 4). find lim f(x) as x→4. enter i for ∞, -i for -∞, and dne if the limit does not exist. answer: submit answer next item
Answer
Explanation:
Step1: Factor the numerator
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 $f(x)=\frac{(x - 4)(x^{2}+4x + 16)}{x - 4}$.
Step2: Simplify the function
Cancel out the common factor $(x - 4)$ for $x\neq4$. So $f(x)=x^{2}+4x + 16$ for $x\neq4$.
Step3: Find the limit
We find $\lim_{x\rightarrow4}f(x)=\lim_{x\rightarrow4}(x^{2}+4x + 16)$. Substitute $x = 4$ into $x^{2}+4x + 16$. We get $4^{2}+4\times4+16=16 + 16+16=48$.
Answer:
48