Calculus Examples

$lim_{h \to 0} \frac{\frac{1}{- 2 + h} + \frac{1}{2}}{h}$

Step 1

Combine terms.

Tap for more steps...

Step 1.1

To write $\frac{1}{- 2 + h}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{h \to 0} \frac{\frac{1}{- 2 + h} \cdot \frac{2}{2} + \frac{1}{2}}{h}$

Step 1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{- 2 + h}{- 2 + h}$ .

$lim_{h \to 0} \frac{\frac{1}{- 2 + h} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{- 2 + h}{- 2 + h}}{h}$

Step 1.3

Write each expression with a common denominator of $(- 2 + h) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.3.1

Multiply $\frac{1}{- 2 + h}$ by $\frac{2}{2}$ .

$lim_{h \to 0} \frac{\frac{2}{(- 2 + h) \cdot 2} + \frac{1}{2} \cdot \frac{- 2 + h}{- 2 + h}}{h}$

Step 1.3.2

Multiply $\frac{1}{2}$ by $\frac{- 2 + h}{- 2 + h}$ .

$lim_{h \to 0} \frac{\frac{2}{(- 2 + h) \cdot 2} + \frac{- 2 + h}{2 (- 2 + h)}}{h}$

Step 1.3.3

Reorder the factors of $(- 2 + h) \cdot 2$ .

$lim_{h \to 0} \frac{\frac{2}{2 (- 2 + h)} + \frac{- 2 + h}{2 (- 2 + h)}}{h}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{2 - 2 + h}{2 (- 2 + h)}}{h}$

Step 1.5

Subtract $2$ from $2$ .

$lim_{h \to 0} \frac{\frac{0 + h}{2 (- 2 + h)}}{h}$

Step 1.6

Add $0$ and $h$ .

$lim_{h \to 0} \frac{\frac{h}{2 (- 2 + h)}}{h}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify the limit argument.

Tap for more steps...

Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{h}{2 (- 2 + h)} \cdot \frac{1}{h}$

Step 2.1.2

Multiply $\frac{h}{2 (- 2 + h)}$ by $\frac{1}{h}$ .

$lim_{h \to 0} \frac{h}{2 (- 2 + h) h}$

Step 2.1.3

Cancel the common factor of $h$ .

Tap for more steps...

Step 2.1.3.1

Cancel the common factor.

$lim_{h \to 0} \frac{h}{2 (- 2 + h) h}$

Step 2.1.3.2

Rewrite the expression.

$lim_{h \to 0} \frac{1}{2 (- 2 + h)}$

Step 2.2

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

$\frac{1}{2} lim_{h \to 0} \frac{1}{- 2 + h}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} - 2 + h}$

Step 2.4

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$\frac{1}{2} \cdot \frac{1}{lim_{h \to 0} - 2 + h}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{1}{2} \cdot \frac{1}{- lim_{h \to 0} 2 + lim_{h \to 0} h}$

Step 2.6

Evaluate the limit of $2$ which is constant as $h$ approaches $0$ .

$\frac{1}{2} \cdot \frac{1}{- 1 \cdot 2 + lim_{h \to 0} h}$

Step 3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{1}{2} \cdot \frac{1}{- 1 \cdot 2 + 0}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Simplify the denominator.

Tap for more steps...

Step 4.1.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} \cdot \frac{1}{- 2 + 0}$

Step 4.1.2

Add $- 2$ and $0$ .

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

Step 4.2

Move the negative in front of the fraction.

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

Step 4.3

Multiply $\frac{1}{2} (- \frac{1}{2})$ .

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Multiply $2$ by $2$ .

$- \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$