Calculus Examples

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

Step 1

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

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

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

Step 2.1.2.1.2

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

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

Step 2.1.2.1.3

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

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

Step 2.1.2.1.4

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

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

Step 2.1.2.1.5

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Add $1$ and $0$ .

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

Step 2.1.2.3.1.2

One to any power is one.

$\frac{1 - 1 \cdot 1}{lim_{h \to 0} h}$

Step 2.1.2.3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{h \to 0} h}$

Step 2.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{h \to 0} h}$

Step 2.1.3

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

$\frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

By the Sum Rule, the derivative of ${(1 + h)}^{3} - 1$ with respect to $h$ is $\frac{d}{d h} [{(1 + h)}^{3}] + \frac{d}{d h} [- 1]$ .

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

Step 2.3.3

Evaluate $\frac{d}{d h} [{(1 + h)}^{3}]$ .

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Step 2.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{3}$ and $g (h) = 1 + h$ .

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Step 2.3.3.1.1

To apply the Chain Rule, set $u$ as $1 + h$ .

$lim_{h \to 0} \frac{\frac{d}{d u} [u^{3}] \frac{d}{d h} [1 + h] + \frac{d}{d h} [- 1]}{\frac{d}{d h} [h]}$

Step 2.3.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$lim_{h \to 0} \frac{3 u^{2} \frac{d}{d h} [1 + h] + \frac{d}{d h} [- 1]}{\frac{d}{d h} [h]}$

Step 2.3.3.1.3

Replace all occurrences of $u$ with $1 + h$ .

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

Step 2.3.3.2

By the Sum Rule, the derivative of $1 + h$ with respect to $h$ is $\frac{d}{d h} [1] + \frac{d}{d h} [h]$ .

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

Step 2.3.3.3

Since $1$ is constant with respect to $h$ , the derivative of $1$ with respect to $h$ is $0$ .

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

Step 2.3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 1$ .

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

Step 2.3.3.5

Add $0$ and $1$ .

$lim_{h \to 0} \frac{3 {(1 + h)}^{2} \cdot 1 + \frac{d}{d h} [- 1]}{\frac{d}{d h} [h]}$

Step 2.3.3.6

Multiply $3$ by $1$ .

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

Step 2.3.4

Since $- 1$ is constant with respect to $h$ , the derivative of $- 1$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{3 {(1 + h)}^{2} + 0}{\frac{d}{d h} [h]}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Add $3 {(1 + h)}^{2}$ and $0$ .

$lim_{h \to 0} \frac{3 {(1 + h)}^{2}}{\frac{d}{d h} [h]}$

Step 2.3.5.2

Rewrite ${(1 + h)}^{2}$ as $(1 + h) (1 + h)$ .

$lim_{h \to 0} \frac{3 ((1 + h) (1 + h))}{\frac{d}{d h} [h]}$

Step 2.3.5.3

Expand $(1 + h) (1 + h)$ using the FOIL Method.

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Step 2.3.5.3.1

Apply the distributive property.

$lim_{h \to 0} \frac{3 (1 (1 + h) + h (1 + h))}{\frac{d}{d h} [h]}$

Step 2.3.5.3.2

Apply the distributive property.

$lim_{h \to 0} \frac{3 (1 \cdot 1 + 1 h + h (1 + h))}{\frac{d}{d h} [h]}$

Step 2.3.5.3.3

Apply the distributive property.

$lim_{h \to 0} \frac{3 (1 \cdot 1 + 1 h + h \cdot 1 + h \cdot h)}{\frac{d}{d h} [h]}$

Step 2.3.5.4

Simplify and combine like terms.

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Step 2.3.5.4.1

Simplify each term.

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Step 2.3.5.4.1.1

Multiply $1$ by $1$ .

$lim_{h \to 0} \frac{3 (1 + 1 h + h \cdot 1 + h \cdot h)}{\frac{d}{d h} [h]}$

Step 2.3.5.4.1.2

Multiply $h$ by $1$ .

$lim_{h \to 0} \frac{3 (1 + h + h \cdot 1 + h \cdot h)}{\frac{d}{d h} [h]}$

Step 2.3.5.4.1.3

Multiply $h$ by $1$ .

$lim_{h \to 0} \frac{3 (1 + h + h + h \cdot h)}{\frac{d}{d h} [h]}$

Step 2.3.5.4.1.4

Multiply $h$ by $h$ .

$lim_{h \to 0} \frac{3 (1 + h + h + h^{2})}{\frac{d}{d h} [h]}$

Step 2.3.5.4.2

Add $h$ and $h$ .

$lim_{h \to 0} \frac{3 (1 + 2 h + h^{2})}{\frac{d}{d h} [h]}$

Step 2.3.5.5

Apply the distributive property.

$lim_{h \to 0} \frac{3 \cdot 1 + 3 (2 h) + 3 h^{2}}{\frac{d}{d h} [h]}$

Step 2.3.5.6

Simplify.

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Step 2.3.5.6.1

Multiply $3$ by $1$ .

$lim_{h \to 0} \frac{3 + 3 (2 h) + 3 h^{2}}{\frac{d}{d h} [h]}$

Step 2.3.5.6.2

Multiply $2$ by $3$ .

$lim_{h \to 0} \frac{3 + 6 h + 3 h^{2}}{\frac{d}{d h} [h]}$

Step 2.3.6

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 1$ .

$lim_{h \to 0} \frac{3 + 6 h + 3 h^{2}}{1}$

Step 2.4

Divide $3 + 6 h + 3 h^{2}$ by $1$ .

$lim_{h \to 0} 3 + 6 h + 3 h^{2}$

Step 3

Evaluate the limit.

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Step 3.1

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

$lim_{h \to 0} 3 + lim_{h \to 0} 6 h + lim_{h \to 0} 3 h^{2}$

Step 3.2

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$3 + lim_{h \to 0} 6 h + lim_{h \to 0} 3 h^{2}$

Step 3.3

Move the term $6$ outside of the limit because it is constant with respect to $h$ .

$3 + 6 lim_{h \to 0} h + lim_{h \to 0} 3 h^{2}$

Step 3.4

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

$3 + 6 lim_{h \to 0} h + 3 lim_{h \to 0} h^{2}$

Step 3.5

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

$3 + 6 lim_{h \to 0} h + 3 {(lim_{h \to 0} h)}^{2}$

Step 4

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

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Step 4.1

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

$3 + 6 \cdot 0 + 3 {(lim_{h \to 0} h)}^{2}$

Step 4.2

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

$3 + 6 \cdot 0 + 3 \cdot 0^{2}$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $6$ by $0$ .

$3 + 0 + 3 \cdot 0^{2}$

Step 5.1.2

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

$3 + 0 + 3 \cdot 0$

Step 5.1.3

Multiply $3$ by $0$ .

$3 + 0 + 0$

Step 5.2

Add $3$ and $0$ .

$3 + 0$

Step 5.3

Add $3$ and $0$ .

$3$