Question

determine if the following limit exists. if it does exist, compute the limit.
lim_{x
ightarrow0}\frac{x^{2}+2x}{x}
if possible, rewrite the limit by simplifying the rational expression. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. (lim_{x
ightarrow0}\frac{x^{2}+2x}{x}=lim_{x
ightarrow0})
b. the limit cannot be rewritten.

Explanation:

Step1: Simplify the rational - expression

Factor out \(x\) from the numerator: \(\lim_{x
ightarrow0}\frac{x^{2}+2x}{x}=\lim_{x
ightarrow0}\frac{x(x + 2)}{x}\). Since \(x
eq0\) when taking the limit (we are approaching 0 but not equal to 0), we can cancel out the \(x\) terms. So \(\lim_{x
ightarrow0}\frac{x(x + 2)}{x}=\lim_{x
ightarrow0}(x + 2)\).

Step2: Evaluate the limit

Substitute \(x = 0\) into the simplified expression \(x+2\). We get \(\lim_{x
ightarrow0}(x + 2)=0 + 2=2\).

Answer:

A. \(\lim_{x
ightarrow0}\frac{x^{2}+2x}{x}=\lim_{x
ightarrow0}(x + 2)\); 2