Calculus Examples

$lim_{h \to 0} 5 x^{3} + 2 x^{2} h - x h^{2}$

Step 1

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

$lim_{h \to 0} 5 x^{3} + lim_{h \to 0} 2 x^{2} h - lim_{h \to 0} x h^{2}$

Step 2

Evaluate the limit of $5 x^{3}$ which is constant as $h$ approaches $0$ .

$5 x^{3} + lim_{h \to 0} 2 x^{2} h - lim_{h \to 0} x h^{2}$

Step 3

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

$5 x^{3} + 2 x^{2} lim_{h \to 0} h - lim_{h \to 0} x h^{2}$

Step 4

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

$5 x^{3} + 2 x^{2} lim_{h \to 0} h - x lim_{h \to 0} h^{2}$

Step 5

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

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

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 $h$ .

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

Step 6.2

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

$5 x^{3} + 2 x^{2} \cdot 0 - x \cdot 0^{2}$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2 x^{2} \cdot 0$ .

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Step 7.1.1.1

Multiply $0$ by $2$ .

$5 x^{3} + 0 x^{2} - x \cdot 0^{2}$

Step 7.1.1.2

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

$5 x^{3} + 0 - x \cdot 0^{2}$

Step 7.1.2

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

$5 x^{3} + 0 - x \cdot 0$

Step 7.1.3

Multiply $- x \cdot 0$ .

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Step 7.1.3.1

Multiply $0$ by $- 1$ .

$5 x^{3} + 0 + 0 x$

Step 7.1.3.2

Multiply $0$ by $x$ .

$5 x^{3} + 0 + 0$

Step 7.2

Combine the opposite terms in $5 x^{3} + 0 + 0$ .

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Step 7.2.1

Add $5 x^{3}$ and $0$ .

$5 x^{3} + 0$

Step 7.2.2

Add $5 x^{3}$ and $0$ .

$5 x^{3}$