determine if the following limit exists. if it does exist, compute the limit. \nlim(x→0) (x³ + 5x)/x\nselect…

determine if the following limit exists. if it does exist, compute the limit. \nlim(x→0) (x³ + 5x)/x\nselect the correct choice below and fill in any answer boxes in your choice.\na. lim(x→0) (x³ + 5x)/x = (simplify your answer.)\nb. the limit does not exist.

determine if the following limit exists. if it does exist, compute the limit. \nlim(x→0) (x³ + 5x)/x\nselect the correct choice below and fill in any answer boxes in your choice.\na. lim(x→0) (x³ + 5x)/x = (simplify your answer.)\nb. the limit does not exist.

Answer

Explanation:

Step1: Simplify the function

Since (x\neq0) when taking the limit as (x\rightarrow0) (we are approaching 0, not setting (x = 0) directly), we can factor out (x) from the numerator and cancel it with the denominator. (\frac{x^{3}+5x}{x}=\frac{x(x^{2} + 5)}{x}=x^{2}+5).

Step2: Substitute (x = 0)

Now we find the limit of (x^{2}+5) as (x\rightarrow0). Substitute (x = 0) into (x^{2}+5), we get (0^{2}+5).

Answer:

A. (\lim_{x\rightarrow0}\frac{x^{3}+5x}{x}=5)