Calculus Examples

$\frac{lim_{h \to 0} 8 {(\frac{1}{2} + h)}^{8} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\frac{lim_{h \to 0} 8 {(\frac{1}{2} + h)}^{8} - lim_{h \to 0} 8 {(\frac{1}{2})}^{8}}{h}$

Step 1.2

Move the term $8$ outside of the limit because it is constant with respect to $h$ .

$\frac{8 lim_{h \to 0} {(\frac{1}{2} + h)}^{8} - lim_{h \to 0} 8 {(\frac{1}{2})}^{8}}{h}$

Step 1.3

Move the exponent $8$ from ${(\frac{1}{2} + h)}^{8}$ outside the limit using the Limits Power Rule.

$\frac{8 {(lim_{h \to 0} \frac{1}{2} + h)}^{8} - lim_{h \to 0} 8 {(\frac{1}{2})}^{8}}{h}$

Step 1.4

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

$\frac{8 {(lim_{h \to 0} \frac{1}{2} + lim_{h \to 0} h)}^{8} - lim_{h \to 0} 8 {(\frac{1}{2})}^{8}}{h}$

Step 1.5

Evaluate the limit of $\frac{1}{2}$ which is constant as $h$ approaches $0$ .

$\frac{8 {(\frac{1}{2} + lim_{h \to 0} h)}^{8} - lim_{h \to 0} 8 {(\frac{1}{2})}^{8}}{h}$

Step 1.6

Evaluate the limit of $8 {(\frac{1}{2})}^{8}$ which is constant as $h$ approaches $0$ .

$\frac{8 {(\frac{1}{2} + lim_{h \to 0} h)}^{8} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 2

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

$\frac{8 {(\frac{1}{2} + 0)}^{8} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Add $\frac{1}{2}$ and $0$ .

$\frac{8 {(\frac{1}{2})}^{8} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{8 \frac{1^{8}}{2^{8}} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.3

One to any power is one.

$\frac{8 \frac{1}{2^{8}} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.4

Raise $2$ to the power of $8$ .

$\frac{8 (\frac{1}{256}) - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.5

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.1.5.1

Factor $8$ out of $256$ .

$\frac{8 \frac{1}{8 (32)} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.5.2

Cancel the common factor.

$\frac{8 \frac{1}{8 \cdot 32} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.5.3

Rewrite the expression.

$\frac{\frac{1}{32} - 8 {(\frac{1}{2})}^{8}}{h}$

Step 3.1.6

Apply the product rule to $\frac{1}{2}$ .

$\frac{\frac{1}{32} - 8 \frac{1^{8}}{2^{8}}}{h}$

Step 3.1.7

One to any power is one.

$\frac{\frac{1}{32} - 8 \frac{1}{2^{8}}}{h}$

Step 3.1.8

Raise $2$ to the power of $8$ .

$\frac{\frac{1}{32} - 8 (\frac{1}{256})}{h}$

Step 3.1.9

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.1.9.1

Factor $8$ out of $- 8$ .

$\frac{\frac{1}{32} + 8 (- 1) \frac{1}{256}}{h}$

Step 3.1.9.2

Factor $8$ out of $256$ .

$\frac{\frac{1}{32} + 8 \cdot - 1 \frac{1}{8 \cdot 32}}{h}$

Step 3.1.9.3

Cancel the common factor.

$\frac{\frac{1}{32} + 8 \cdot - 1 \frac{1}{8 \cdot 32}}{h}$

Step 3.1.9.4

Rewrite the expression.

$\frac{\frac{1}{32} - 1 (\frac{1}{32})}{h}$

Step 3.1.10

Rewrite $- 1 (\frac{1}{32})$ as $- (\frac{1}{32})$ .

$\frac{\frac{1}{32} - (\frac{1}{32})}{h}$

Step 3.1.11

Combine the numerators over the common denominator.

$\frac{\frac{1 - 1}{32}}{h}$

Step 3.1.12

Subtract $1$ from $1$ .

$\frac{\frac{0}{32}}{h}$

Step 3.1.13

Divide $0$ by $32$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$