Question

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

Explanation:

Step1: Simplify the function

Since \(x
eq0\) when taking the limit as \(x
ightarrow0\) (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 \(x
ightarrow0\). Substitute \(x = 0\) into \(x^{2}+5\), we get \(0^{2}+5\).

Answer:

A. \(\lim_{x
ightarrow0}\frac{x^{3}+5x}{x}=5\)