Question

Find the limit (Hint: compute the one-sided limits first)
lim|x+2|(x+3)/x+2
x->-2

Answer 1

\[\lim_{x\to-2}\frac{|x+2|(x+3)}{x+2}\]
By definition of absolute value,
\[|x+2|=\begin{cases}x+2&\text{if $x+2\ge0$}\\-(x+2)&\text{if $x+2<0$}\end{cases}\]
This means that for values of x less than -2, and thus as x approaches -2 from the left,
\[|x+2|=-(x+2), \text{so}\\
\lim_{x\to-2^-}\frac{-(x+2)(x+3)}{x+2}\]
Additionally, for values of x greater than -2, and thus as x approaches -2 from the right,
\[|x+2|=x+2, \text{so}\\
\lim_{x\to-2^+}\frac{(x+2)(x+3)}{x+2}\]
Now you can easily determine the limits.