Calculus Examples

$lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{2}}{x - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} \frac{1}{x} - \frac{1}{2}}{lim_{x \to 3} x - 2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} \frac{1}{x} - lim_{x \to 3} \frac{1}{2}}{lim_{x \to 3} x - 2}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{\frac{lim_{x \to 3} 1}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{2}}{lim_{x \to 3} x - 2}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\frac{\frac{1}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{2}}{lim_{x \to 3} x - 2}$

Step 5

Evaluate the limit of $\frac{1}{2}$ which is constant as $x$ approaches $3$ .

$\frac{\frac{1}{lim_{x \to 3} x} - \frac{1}{2}}{lim_{x \to 3} x - 2}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{\frac{1}{lim_{x \to 3} x} - \frac{1}{2}}{lim_{x \to 3} x - lim_{x \to 3} 2}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $3$ .

$\frac{\frac{1}{lim_{x \to 3} x} - \frac{1}{2}}{lim_{x \to 3} x - 1 \cdot 2}$

Step 8

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{\frac{1}{3} - \frac{1}{2}}{lim_{x \to 3} x - 1 \cdot 2}$

Step 8.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{\frac{1}{3} - \frac{1}{2}}{3 - 1 \cdot 2}$

Step 9

Simplify the answer.

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Step 9.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 9.1.1

Multiply $\frac{\frac{1}{3} - \frac{1}{2}}{3 - 1 \cdot 2}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\frac{1}{3} - \frac{1}{2}}{3 - 1 \cdot 2}$

Step 9.1.2

Combine.

$\frac{6 (\frac{1}{3} - \frac{1}{2})}{6 (3 - 1 \cdot 2)}$

Step 9.2

Apply the distributive property.

$\frac{6 (\frac{1}{3}) + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3

Simplify by cancelling.

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Step 9.3.1

Cancel the common factor of $3$ .

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Step 9.3.1.1

Factor $3$ out of $6$ .

$\frac{3 (2) \frac{1}{3} + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.1.2

Cancel the common factor.

$\frac{3 \cdot 2 \frac{1}{3} + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.1.3

Rewrite the expression.

$\frac{2 + 6 (- \frac{1}{2})}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.2

Cancel the common factor of $2$ .

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Step 9.3.2.1

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

$\frac{2 + 6 (\frac{- 1}{2})}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.2.2

Factor $2$ out of $6$ .

$\frac{2 + 2 (3) \frac{- 1}{2}}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.2.3

Cancel the common factor.

$\frac{2 + 2 \cdot 3 \frac{- 1}{2}}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.2.4

Rewrite the expression.

$\frac{2 + 3 \cdot - 1}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.3.3

Multiply $3$ by $- 1$ .

$\frac{2 - 3}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.4

Subtract $3$ from $2$ .

$\frac{- 1}{6 \cdot 3 + 6 (- 1 \cdot 2)}$

Step 9.5

Simplify the denominator.

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Step 9.5.1

Multiply $6$ by $3$ .

$\frac{- 1}{18 + 6 (- 1 \cdot 2)}$

Step 9.5.2

Multiply $6 (- 1 \cdot 2)$ .

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Step 9.5.2.1

Multiply $- 1$ by $2$ .

$\frac{- 1}{18 + 6 \cdot - 2}$

Step 9.5.2.2

Multiply $6$ by $- 2$ .

$\frac{- 1}{18 - 12}$

Step 9.5.3

Subtract $12$ from $18$ .

$\frac{- 1}{6}$

Step 9.6

Move the negative in front of the fraction.

$- \frac{1}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{6}$

Decimal Form:

$- 0.1\overline{6}$