Question

instructions: please show all your work. the method of solution, the final answer to how you got it. no books, notes, or aids. even if you have had calculus before, you will not be given any credit.
q) (10 points) evaluate the limit.
lim_{x
ightarrow - 2}\frac{3x^{2}+6x}{|x + 2|}

Explanation:

Step1: Analyze the left - hand limit

As \(x\to2^{-}\), \(x + 2>0\), so \(|x + 2|=x + 2\).

Step2: Substitute into the limit expression

We have \(\lim_{x\to2^{-}}\frac{3x^{2}+6x}{|x + 2|}=\lim_{x\to2^{-}}\frac{3x(x + 2)}{x + 2}\).

Step3: Simplify the expression

Cancel out the common factor \((x + 2)\) (since \(x
eq - 2\) when \(x\to2^{-}\)), we get \(\lim_{x\to2^{-}}3x\).

Step4: Evaluate the limit

Substitute \(x = 2\) into \(3x\), we have \(3\times2=6\).

Answer:

6