Calculus Examples

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

Step 1

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

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

Step 2

Simplify the answer.

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Step 2.1

Multiply the numerator and denominator of the fraction by $y \cdot 2$ .

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Step 2.1.1

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

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

Step 2.1.2

Combine.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Simplify by cancelling.

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Step 2.3.1

Cancel the common factor of $y$ .

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Step 2.3.1.1

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

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

Step 2.3.1.2

Cancel the common factor.

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

Step 2.3.1.3

Rewrite the expression.

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

Cancel the common factor.

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

Step 2.3.2.4

Rewrite the expression.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Move $- 1$ to the left of $y$ .

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

Step 2.4.2

Rewrite $- 1 y$ as $- y$ .

$\frac{2 - y}{y \cdot 2 y + y \cdot 2 \cdot - 2}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Multiply $y$ by $y$ by adding the exponents.

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Step 2.5.1.1

Move $y$ .

$\frac{2 - y}{y \cdot y \cdot 2 + y \cdot 2 \cdot - 2}$

Step 2.5.1.2

Multiply $y$ by $y$ .

$\frac{2 - y}{y^{2} \cdot 2 + y \cdot 2 \cdot - 2}$

Step 2.5.2

Move $2$ to the left of $y^{2}$ .

$\frac{2 - y}{2 \cdot y^{2} + y \cdot 2 \cdot - 2}$

Step 2.5.3

Move $2$ to the left of $y$ .

$\frac{2 - y}{2 y^{2} + 2 \cdot y \cdot - 2}$

Step 2.5.4

Multiply $- 2$ by $2$ .

$\frac{2 - y}{2 y^{2} - 4 y}$

Step 2.5.5

Factor $2 y$ out of $2 y^{2} - 4 y$ .

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Step 2.5.5.1

Factor $2 y$ out of $2 y^{2}$ .

$\frac{2 - y}{2 y (y) - 4 y}$

Step 2.5.5.2

Factor $2 y$ out of $- 4 y$ .

$\frac{2 - y}{2 y (y) + 2 y (- 2)}$

Step 2.5.5.3

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

$\frac{2 - y}{2 y (y - 2)}$

Step 2.6

Cancel the common factor of $2 - y$ and $y - 2$ .

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Step 2.6.1

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 2.6.2

Factor $- 1$ out of $- y$ .

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

Step 2.6.3

Factor $- 1$ out of $- 1 (- 2) - (y)$ .

$\frac{- 1 (- 2 + y)}{2 y (y - 2)}$

Step 2.6.4

Reorder terms.

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

Step 2.6.5

Cancel the common factor.

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

Step 2.6.6

Rewrite the expression.

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

Step 2.7

Move the negative in front of the fraction.

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