Calculus Examples

$lim_{x \to 2} - \frac{\frac{x - 2}{2 x}}{x - 2}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 2} \frac{\frac{x - 2}{2 x}}{x - 2}$

Step 1.2

Simplify the limit argument.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$- lim_{x \to 2} \frac{x - 2}{2 x} \cdot \frac{1}{x - 2}$

Step 1.2.2

Multiply $\frac{x - 2}{2 x}$ by $\frac{1}{x - 2}$ .

$- lim_{x \to 2} \frac{x - 2}{2 x (x - 2)}$

Step 1.2.3

Cancel the common factor of $x - 2$ .

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Step 1.2.3.1

Cancel the common factor.

$- lim_{x \to 2} \frac{x - 2}{2 x (x - 2)}$

Step 1.2.3.2

Rewrite the expression.

$- lim_{x \to 2} \frac{1}{2 x}$

Step 1.3

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{2} lim_{x \to 2} \frac{1}{x}$

Step 1.4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$- \frac{1}{2} \cdot \frac{lim_{x \to 2} 1}{lim_{x \to 2} x}$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$- \frac{1}{2} \cdot \frac{1}{lim_{x \to 2} x}$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$- \frac{1}{2} \cdot \frac{1}{2}$

Step 3

Multiply $- \frac{1}{2} \cdot \frac{1}{2}$ .

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Step 3.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$- \frac{1}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$- \frac{1}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$