Calculus Examples

$\frac{lim_{h \to 0} {(- x + h)}^{3} + 13 {(x + h)}^{2} - 40 (x + h) - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1

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

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Step 1.1

Evaluate the limit of ${(- x + h)}^{3} + 13 {(x + h)}^{2} - 40 (x + h) - (- x^{3} + 13 x^{2} - 40 x)$ by plugging in $0$ for $h$ .

$\frac{{(- x + 0)}^{3} + 13 {(x + 0)}^{2} - 40 (x + 0) - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1.2

Combine the opposite terms in ${(- x + 0)}^{3} + 13 {(x + 0)}^{2} - 40 (x + 0) - (- x^{3} + 13 x^{2} - 40 x)$ .

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Step 1.2.1

Add $- x$ and $0$ .

$\frac{{(- x)}^{3} + 13 {(x + 0)}^{2} - 40 (x + 0) - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1.2.2

Add $x$ and $0$ .

$\frac{{(- x)}^{3} + 13 x^{2} - 40 (x + 0) - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1.2.3

Add $x$ and $0$ .

$\frac{{(- x)}^{3} + 13 x^{2} - 40 x - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the product rule to $- x$ .

$\frac{{(- 1)}^{3} x^{3} + 13 x^{2} - 40 x - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1.3.2

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

$\frac{- x^{3} + 13 x^{2} - 40 x - (- x^{3} + 13 x^{2} - 40 x)}{h^{2}}$

Step 1.3.3

Apply the distributive property.

$\frac{- x^{3} + 13 x^{2} - 40 x - - x^{3} - (13 x^{2}) - (- 40 x)}{h^{2}}$

Step 1.3.4

Simplify.

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Step 1.3.4.1

Multiply $- - x^{3}$ .

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Step 1.3.4.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- x^{3} + 13 x^{2} - 40 x + 1 x^{3} - (13 x^{2}) - (- 40 x)}{h^{2}}$

Step 1.3.4.1.2

Multiply $x^{3}$ by $1$ .

$\frac{- x^{3} + 13 x^{2} - 40 x + x^{3} - (13 x^{2}) - (- 40 x)}{h^{2}}$

Step 1.3.4.2

Multiply $13$ by $- 1$ .

$\frac{- x^{3} + 13 x^{2} - 40 x + x^{3} - 13 x^{2} - (- 40 x)}{h^{2}}$

Step 1.3.4.3

Multiply $- 40$ by $- 1$ .

$\frac{- x^{3} + 13 x^{2} - 40 x + x^{3} - 13 x^{2} + 40 x}{h^{2}}$

Step 1.4

Combine the opposite terms in $- x^{3} + 13 x^{2} - 40 x + x^{3} - 13 x^{2} + 40 x$ .

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Step 1.4.1

Add $- x^{3}$ and $x^{3}$ .

$\frac{13 x^{2} - 40 x + 0 - 13 x^{2} + 40 x}{h^{2}}$

Step 1.4.2

Add $13 x^{2} - 40 x$ and $0$ .

$\frac{13 x^{2} - 40 x - 13 x^{2} + 40 x}{h^{2}}$

Step 1.4.3

Subtract $13 x^{2}$ from $13 x^{2}$ .

$\frac{- 40 x + 0 + 40 x}{h^{2}}$

Step 1.4.4

Add $- 40 x$ and $0$ .

$\frac{- 40 x + 40 x}{h^{2}}$

Step 1.4.5

Add $- 40 x$ and $40 x$ .

$\frac{0}{h^{2}}$

Step 2

Divide $0$ by $h^{2}$ .

$0$