Calculus Examples

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

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 $h$ approaches $0$ .

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

Step 1.2

Move the exponent $3$ from ${(h - 1)}^{3}$ outside the limit using the Limits Power Rule.

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $1$ .

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

Step 3.1.2

Subtract $1$ from $0$ .

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

Step 3.1.3

Raise $- 1$ to the power of $3$ .

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

Step 3.1.4

Add $- 1$ and $1$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$