Calculus Examples

$\frac{lim_{h \to 0} (3 {(\frac{1}{2} + h)}^{5}) - 3 {(\frac{1}{2})}^{5}}{h}$

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 $0$ .

$\frac{lim_{h \to 0} 3 {(\frac{1}{2} + h)}^{5} - lim_{h \to 0} 3 {(\frac{1}{2})}^{5}}{h}$

Step 1.2

Move the term $3$ outside of the limit because it is constant with respect to $h$ .

$\frac{3 lim_{h \to 0} {(\frac{1}{2} + h)}^{5} - lim_{h \to 0} 3 {(\frac{1}{2})}^{5}}{h}$

Step 1.3

Move the exponent $5$ from ${(\frac{1}{2} + h)}^{5}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{h \to 0} \frac{1}{2} + h)}^{5} - lim_{h \to 0} 3 {(\frac{1}{2})}^{5}}{h}$

Step 1.4

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

$\frac{3 {(lim_{h \to 0} \frac{1}{2} + lim_{h \to 0} h)}^{5} - lim_{h \to 0} 3 {(\frac{1}{2})}^{5}}{h}$

Step 1.5

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

$\frac{3 {(\frac{1}{2} + lim_{h \to 0} h)}^{5} - lim_{h \to 0} 3 {(\frac{1}{2})}^{5}}{h}$

Step 1.6

Evaluate the limit of $3 {(\frac{1}{2})}^{5}$ which is constant as $h$ approaches $0$ .

$\frac{3 {(\frac{1}{2} + lim_{h \to 0} h)}^{5} - 3 {(\frac{1}{2})}^{5}}{h}$

Step 2

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

$\frac{3 {(\frac{1}{2} + 0)}^{5} - 3 {(\frac{1}{2})}^{5}}{h}$

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 $0$ .

$\frac{3 {(\frac{1}{2})}^{5} - 3 {(\frac{1}{2})}^{5}}{h}$

Step 3.1.2

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

$\frac{3 \frac{1^{5}}{2^{5}} - 3 {(\frac{1}{2})}^{5}}{h}$

Step 3.1.3

One to any power is one.

$\frac{3 \frac{1}{2^{5}} - 3 {(\frac{1}{2})}^{5}}{h}$

Step 3.1.4

Raise $2$ to the power of $5$ .

$\frac{3 (\frac{1}{32}) - 3 {(\frac{1}{2})}^{5}}{h}$

Step 3.1.5

Combine $3$ and $\frac{1}{32}$ .

$\frac{\frac{3}{32} - 3 {(\frac{1}{2})}^{5}}{h}$

Step 3.1.6

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

$\frac{\frac{3}{32} - 3 \frac{1^{5}}{2^{5}}}{h}$

Step 3.1.7

One to any power is one.

$\frac{\frac{3}{32} - 3 \frac{1}{2^{5}}}{h}$

Step 3.1.8

Raise $2$ to the power of $5$ .

$\frac{\frac{3}{32} - 3 (\frac{1}{32})}{h}$

Step 3.1.9

Combine $- 3$ and $\frac{1}{32}$ .

$\frac{\frac{3}{32} + \frac{- 3}{32}}{h}$

Step 3.1.10

Move the negative in front of the fraction.

$\frac{\frac{3}{32} - \frac{3}{32}}{h}$

Step 3.1.11

Combine the numerators over the common denominator.

$\frac{\frac{3 - 3}{32}}{h}$

Step 3.1.12

Subtract $3$ from $3$ .

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

Step 3.1.13

Divide $0$ by $32$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$