Calculus Examples

$\frac{lim_{x \to 5} \frac{2}{x - 1} - 2}{x - 2}$

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 $5$ .

$\frac{lim_{x \to 5} \frac{2}{x - 1} - lim_{x \to 5} 2}{x - 2}$

Step 1.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 5} \frac{1}{x - 1} - lim_{x \to 5} 2}{x - 2}$

Step 1.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{2 \frac{lim_{x \to 5} 1}{lim_{x \to 5} x - 1} - lim_{x \to 5} 2}{x - 2}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\frac{2 \frac{1}{lim_{x \to 5} x - 1} - lim_{x \to 5} 2}{x - 2}$

Step 1.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{2 \frac{1}{lim_{x \to 5} x - lim_{x \to 5} 1} - lim_{x \to 5} 2}{x - 2}$

Step 1.6

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\frac{2 \frac{1}{lim_{x \to 5} x - 1 \cdot 1} - lim_{x \to 5} 2}{x - 2}$

Step 1.7

Evaluate the limit of $2$ which is constant as $x$ approaches $5$ .

$\frac{2 \frac{1}{lim_{x \to 5} x - 1 \cdot 1} - 1 \cdot 2}{x - 2}$

Step 2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{2 \frac{1}{5 - 1 \cdot 1} - 1 \cdot 2}{x - 2}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Multiply $- 1$ by $1$ .

$\frac{2 \frac{1}{5 - 1} - 1 \cdot 2}{x - 2}$

Step 3.1.2

Subtract $1$ from $5$ .

$\frac{2 (\frac{1}{4}) - 1 \cdot 2}{x - 2}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Factor $2$ out of $4$ .

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

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

Step 3.2.2

Multiply $- 1$ by $2$ .

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

Step 3.2.3

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.4

Combine $- 2$ and $\frac{2}{2}$ .

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 2$ by $2$ .

$\frac{\frac{1 - 4}{2}}{x - 2}$

Step 3.2.6.2

Subtract $4$ from $1$ .

$\frac{\frac{- 3}{2}}{x - 2}$

Step 3.2.7

Move the negative in front of the fraction.

$\frac{- \frac{3}{2}}{x - 2}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.4

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

$- \frac{3}{(x - 2) \cdot 2}$

Step 3.5

Move $2$ to the left of $x - 2$ .

$- \frac{3}{2 (x - 2)}$