Calculus Examples

$\frac{lim_{h \to 0} x^{2} h - x h^{2} + h^{3}}{h}$

Step 1

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

$\frac{lim_{h \to 0} x^{2} h - lim_{h \to 0} x h^{2} + lim_{h \to 0} h^{3}}{h}$

Step 2

Move the term $x^{2}$ outside of the limit because it is constant with respect to $h$ .

$\frac{x^{2} lim_{h \to 0} h - lim_{h \to 0} x h^{2} + lim_{h \to 0} h^{3}}{h}$

Step 3

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

$\frac{x^{2} lim_{h \to 0} h - x lim_{h \to 0} h^{2} + lim_{h \to 0} h^{3}}{h}$

Step 4

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

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

Step 5

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

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

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

$\frac{x^{2} \cdot 0 - x \cdot 0^{2} + 0^{3}}{h}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $x^{2}$ by $0$ .

$\frac{0 - x \cdot 0^{2} + 0^{3}}{h}$

Step 7.1.2

Raising $0$ to any positive power yields $0$ .

$\frac{0 - x \cdot 0 + 0^{3}}{h}$

Step 7.1.3

Multiply $- x \cdot 0$ .

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Step 7.1.3.1

Multiply $0$ by $- 1$ .

$\frac{0 + 0 x + 0^{3}}{h}$

Step 7.1.3.2

Multiply $0$ by $x$ .

$\frac{0 + 0 + 0^{3}}{h}$

Step 7.1.4

Raising $0$ to any positive power yields $0$ .

$\frac{0 + 0 + 0}{h}$

Step 7.1.5

Add $0 + 0$ and $0$ .

$\frac{0 + 0}{h}$

Step 7.1.6

Add $0$ and $0$ .

$\frac{0}{h}$

Step 7.2

Divide $0$ by $h$ .

$0$