Question

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

47. \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{0}}^ + }} \left( {\frac{{\bf{1}}}{x} - \frac{{\bf{1}}}{{\left| x \right|}}} \right)\)

Short answer

Limit does not exist.

Step-by-step solution

Simplify the expression given in the limit

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

The expression in the limit can be written as,

\(\begin{array}{c}\mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{1}{x} - \frac{1}{{\left| x \right|}}} \right) &=& \mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{1}{x} - \left( { - \frac{1}{x}} \right)} \right)\\ &=& \mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{2}{x}} \right)\end{array}\)

Find the value of the limit

For the expression \(\mathop {\lim }\limits_{x \to {0^ - }} \left( {\frac{2}{x}} \right)\), as \(x \to 0\)the term \(\frac{2}{x}\) is not defined.

So, the limit does not exist.