Question

Evaluate the limit, if it exists.

\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} \frac{{{\bf{2}} - x}}{{\sqrt {x + {\bf{2}}} - {\bf{2}}}}\)

Short answer

The value of the limit is \( - 4\).

Step-by-step solution

Simplify the limit by using the factorization

Rationalize the expression \(\mathop {\lim }\limits_{x \to 2} \frac{{2 - x}}{{\sqrt {x + 2} - 2}}\).

\(\begin{aligned}\mathop {\lim }\limits_{x \to 2} \frac{{2 - x}}{{\sqrt {x + 2} - 2}} &= \mathop {\lim }\limits_{x \to 2} \frac{{2 - x}}{{\sqrt {x + 2} - 2}} * \frac{{\sqrt {x + 2} + 2}}{{\sqrt {x + 2} + 2}}\\ &= \mathop {\lim }\limits_{x \to 2} \frac{{\left( {2 - x} \right)\left( {\sqrt {x + 2} + 2} \right)}}{{{{\left( {\sqrt {x + 2} } \right)}^2} - 4}}\\ &= \mathop {\lim }\limits_{x \to 2} \frac{{ - \left( {x - 2} \right)\left( {\sqrt {x + 2} + 2} \right)}}{{\left( {x - 2} \right)}}\\ &= \mathop {\lim }\limits_{x \to 2} \left( { - \left( {\sqrt {x + 2} + 2} \right)} \right)\end{aligned}\)

Evaluate the limit

For the expression,\(\mathop {\lim }\limits_{x \to 2} \left( { - \left( {\sqrt {x + 2} + 2} \right)} \right)\) Difference law is applicable.

According to the difference law, \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - g\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) - \mathop {\lim }\limits_{x \to a} g\left( x \right)\), where \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) and \(\mathop {\lim }\limits_{x \to a} g\left( x \right)\) exists.

Solve the expression \(\mathop {\lim }\limits_{x \to 2} \left( { - \left( {\sqrt {x + 2} + 2} \right)} \right)\).

\(\begin{aligned}\mathop {\lim }\limits_{x \to 2} \left( { - \left( {\sqrt {x + 2} + 2} \right)} \right) &= \mathop {\lim }\limits_{x \to 2} \left( { - \sqrt {x + 2} } \right) + \mathop {\lim }\limits_{x \to 2} \left( 2 \right)\\ &= - \left( {\sqrt {2 + 2} + 2} \right)\\ &= - 4\end{aligned}\)

Thus, the value of the limit is \( - 4\).