Question

43-48 Find the limit, if it exists. If the limit does not exist, explain why.

46. \(\mathop {{\bf{lim}}}\limits_{x \to - 2} \frac{{{\bf{2}} - \left| x \right|}}{{{\bf{2}} + x}}\)

Short answer

Thus, the required answer is 1.

Step-by-step solution

Simplify the expression given in the limit

If \(x = - 2\) (\(x < 0\)), then \(\left| x \right| = - x\).

The expression in the limit can be written as,

\(\mathop {\lim }\limits_{x \to - 2} \frac{{2 - \left( { - x} \right)}}{{2 + x}} = \mathop {\lim }\limits_{x \to - 2} \frac{{2 + x}}{{2 + x}}\)

Find the value of limit

The value of the limit can be calculated as,

\(\begin{array}{c}\mathop {\lim }\limits_{x \to - 2} \frac{{2 + x}}{{2 + x}} &=& \mathop {\lim }\limits_{x \to - 2} \left( 1 \right)\\ &=& 1\end{array}\)

So, the value of the limit is 1.