Question

determine the following limit, using \\(\infty\\) or \\(-\infty\\) when appropriate, or state that it does not exist. \\(\lim_{x \to 0} \frac{x^3 + 4x^2}{x^2}\\) simplify the expression inside the limit, if possible. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. \\(\lim_{x \to 0} \frac{x^3 + 4x^2}{x^2} = \lim_{x \to 0} \left( \square \
ight)\\) b. the expression inside the limit cannot be simplified.

Explanation:

Step1: Factor numerator

Factor \(x^2\) from \(x^3 + 4x^2\): \(x^2(x + 4)\)

Step2: Simplify fraction

Divide \(\frac{x^2(x + 4)}{x^2}\) (for \(x
eq0\), \(x^2
eq0\)): \(x + 4\)

Step3: Rewrite limit

So \(\lim_{x\to0}\frac{x^3 + 4x^2}{x^2}=\lim_{x\to0}(x + 4)\)

Answer:

A. \(\lim\limits_{x\to0}\frac{x^3 + 4x^2}{x^2}=\lim\limits_{x\to0}\boldsymbol{(x + 4)}\)