Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Evaluate the limit of $\frac{1}{2}$ which is constant as $x$ approaches $2$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine.

$\frac{2 (\frac{1}{2} - \frac{1}{2})}{2 (x - 2)}$

Step 3.2

Apply the distributive property.

$\frac{2 (\frac{1}{2}) + 2 (- \frac{1}{2})}{2 x + 2 \cdot - 2}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{2 (\frac{1}{2}) + 2 (- \frac{1}{2})}{2 x + 2 \cdot - 2}$

Step 3.3.1.2

Rewrite the expression.

$\frac{1 + 2 (- \frac{1}{2})}{2 x + 2 \cdot - 2}$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{1 + 2 (\frac{- 1}{2})}{2 x + 2 \cdot - 2}$

Step 3.3.2.2

Cancel the common factor.

$\frac{1 + 2 (\frac{- 1}{2})}{2 x + 2 \cdot - 2}$

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Subtract $1$ from $1$ .

$\frac{0}{2 x + 2 \cdot - 2}$

Step 3.5

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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Step 3.5.1

Factor $2$ out of $2 x$ .

$\frac{0}{2 (x) + 2 \cdot - 2}$

Step 3.5.2

Factor $2$ out of $2 \cdot - 2$ .

$\frac{0}{2 (x) + 2 (- 2)}$

Step 3.5.3

Factor $2$ out of $2 (x) + 2 (- 2)$ .

$\frac{0}{2 (x - 2)}$

Step 3.6

Cancel the common factor of $0$ and $2$ .

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Step 3.6.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 (x - 2)}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Cancel the common factor.

$\frac{2 \cdot 0}{2 (x - 2)}$

Step 3.6.2.2

Rewrite the expression.

$\frac{0}{x - 2}$

Step 3.7

Divide $0$ by $x - 2$ .

$0$