Calculus Examples

lim x \to 0 (\frac{1}{x}) - \frac{1}{| x |}

$lim_{x \to 0} (\frac{1}{x}) - \frac{1}{| x |}$

Step 1

Combine terms.

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Step 1.1

To write

\frac{1}{x}

$\frac{1}{x}$ as a fraction with a common denominator, multiply by

\frac{| x |}{| x |}

$\frac{| x |}{| x |}$ .

lim x \to 0 \frac{1}{x} \cdot \frac{| x |}{| x |} - \frac{1}{| x |}

$lim_{x \to 0} \frac{1}{x} \cdot \frac{| x |}{| x |} - \frac{1}{| x |}$

Step 1.2

To write

- \frac{1}{| x |}

$- \frac{1}{| x |}$ as a fraction with a common denominator, multiply by

\frac{x}{x}

$\frac{x}{x}$ .

lim x \to 0 \frac{1}{x} \cdot \frac{| x |}{| x |} - \frac{1}{| x |} \cdot \frac{x}{x}

$lim_{x \to 0} \frac{1}{x} \cdot \frac{| x |}{| x |} - \frac{1}{| x |} \cdot \frac{x}{x}$

Step 1.3

Write each expression with a common denominator of

x | x |

$x | x |$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.3.1

Multiply

\frac{1}{x}

$\frac{1}{x}$ by

\frac{| x |}{| x |}

$\frac{| x |}{| x |}$ .

lim x \to 0 \frac{| x |}{x | x |} - \frac{1}{| x |} \cdot \frac{x}{x}

$lim_{x \to 0} \frac{| x |}{x | x |} - \frac{1}{| x |} \cdot \frac{x}{x}$

Step 1.3.2

Multiply

\frac{1}{| x |}

$\frac{1}{| x |}$ by

\frac{x}{x}

$\frac{x}{x}$ .

$lim_{x \to 0} \frac{| x |}{x | x |} - \frac{x}{| x | x}$

Step 1.3.3

Reorder the factors of

$| x | x$ .

$lim_{x \to 0} \frac{| x |}{x | x |} - \frac{x}{x | x |}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{| x | - x}{x | x |}$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \frac{| x | - x}{x | x |}$

Step 3

As the

$x$ values approach

$0$ from the left, the function values decrease without bound.

$- \infty$

Step 4

Consider the right sided limit.

$lim_{x \to 0^{+}} \frac{| x | - x}{x | x |}$

Step 5

Make a table to show the behavior of the function

$\frac{| x | - x}{x | x |}$ as

$x$ approaches

$0$ from the right.

$\begin{array}{c} x & \frac{| x | - x}{x | x |} \\ 0.1 & 0 \\ 0.01 & 0 \\ 0.001 & 0 \end{array}$

Step 6

As the

$x$ values approach

$0$ , the function values approach

$0$ . Thus, the limit of

$\frac{| x | - x}{x | x |}$ as

$x$ approaches

$0$ from the right is

$0$ .

$0$

Step 7

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$