determine if the following limit exists. if it does exist, compute the limit. lim(x→0) (x^4 + 6x)/x select…

Explanation:

Step1: Simplify the function

Divide each term in the numerator by $x$. So, $\frac{x^{4}+6x}{x}=\frac{x^{4}}{x}+\frac{6x}{x}=x^{3} + 6$.

Step2: Compute the limit

Use the limit - rule $\lim_{x\rightarrow a}(f(x)+g(x))=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$ and $\lim_{x\rightarrow a}x^{n}=a^{n}$. Here, $\lim_{x\rightarrow0}(x^{3}+6)=\lim_{x\rightarrow0}x^{3}+\lim_{x\rightarrow0}6$. Since $\lim_{x\rightarrow0}x^{3}=0^{3} = 0$ and $\lim_{x\rightarrow0}6 = 6$, then $\lim_{x\rightarrow0}(x^{3}+6)=0 + 6=6$.

Answer:

A. $\lim_{x\rightarrow0}\frac{x^{4}+6x}{x}=6$