Calculus Examples

$lim_{h \to 0} \frac{5 {(a - h)}^{2} - 5 a^{2}}{h}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{h \to 0} 5 {(a - h)}^{2} - 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{h \to 0} 5 {(a - h)}^{2} - lim_{h \to 0} 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.2

Move the term $5$ outside of the limit because it is constant with respect to $h$ .

$\frac{5 lim_{h \to 0} {(a - h)}^{2} - lim_{h \to 0} 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3

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

$\frac{5 {(lim_{h \to 0} a - h)}^{2} - lim_{h \to 0} 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.4

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

$\frac{5 {(lim_{h \to 0} a - lim_{h \to 0} h)}^{2} - lim_{h \to 0} 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.5

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

$\frac{5 {(a - lim_{h \to 0} h)}^{2} - lim_{h \to 0} 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.6

Evaluate the limit of $5 a^{2}$ which is constant as $h$ approaches $0$ .

$\frac{5 {(a - lim_{h \to 0} h)}^{2} - 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.7

Simplify by adding terms.

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Step 1.1.2.7.1

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

$\frac{5 {(a + 0)}^{2} - 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.7.2

Combine the opposite terms in $5 {(a + 0)}^{2} - 5 a^{2}$ .

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Step 1.1.2.7.2.1

Add $a$ and $0$ .

$\frac{5 a^{2} - 5 a^{2}}{lim_{h \to 0} h}$

Step 1.1.2.7.2.2

Subtract $5 a^{2}$ from $5 a^{2}$ .

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

Step 1.1.3

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

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{h \to 0} \frac{5 {(a - h)}^{2} - 5 a^{2}}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [5 {(a - h)}^{2} - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 {(a - h)}^{2} - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.2

Rewrite ${(a - h)}^{2}$ as $(a - h) (a - h)$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 ((a - h) (a - h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.3

Expand $(a - h) (a - h)$ using the FOIL Method.

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Step 1.3.3.1

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a (a - h) - h (a - h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.3.2

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a \cdot a + a (- h) - h (a - h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.3.3

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a \cdot a + a (- h) - h a - h (- h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $a$ by $a$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} + a (- h) - h a - h (- h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.1.2

Rewrite using the commutative property of multiplication.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - h a - h (- h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.1.3

Rewrite using the commutative property of multiplication.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - h a - 1 \cdot - 1 h \cdot h) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.1.4

Multiply $h$ by $h$ by adding the exponents.

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Step 1.3.4.1.4.1

Move $h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - h a - 1 \cdot - 1 (h \cdot h)) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.1.4.2

Multiply $h$ by $h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - h a - 1 \cdot - 1 h^{2}) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.1.5

Multiply $- 1$ by $- 1$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - h a + 1 h^{2}) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.1.6

Multiply $h^{2}$ by $1$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - h a + h^{2}) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.2

Subtract $h a$ from $- a h$ .

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Step 1.3.4.2.1

Move $h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - a h - 1 a h + h^{2}) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.2.2

Subtract $a h$ from $- a h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - 2 a h + h^{2}) - 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.5

By the Sum Rule, the derivative of $5 (a^{2} - 2 a h + h^{2}) - 5 a^{2}$ with respect to $h$ is $\frac{d}{d h} [5 (a^{2} - 2 a h + h^{2})] + \frac{d}{d h} [- 5 a^{2}]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 (a^{2} - 2 a h + h^{2})] + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6

Evaluate $\frac{d}{d h} [5 (a^{2} - 2 a h + h^{2})]$ .

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Step 1.3.6.1

Since $5$ is constant with respect to $h$ , the derivative of $5 (a^{2} - 2 a h + h^{2})$ with respect to $h$ is $5 \frac{d}{d h} [a^{2} - 2 a h + h^{2}]$ .

$lim_{h \to 0} \frac{5 \frac{d}{d h} [a^{2} - 2 a h + h^{2}] + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.2

By the Sum Rule, the derivative of $a^{2} - 2 a h + h^{2}$ with respect to $h$ is $\frac{d}{d h} [a^{2}] + \frac{d}{d h} [- 2 a h] + \frac{d}{d h} [h^{2}]$ .

$lim_{h \to 0} \frac{5 (\frac{d}{d h} [a^{2}] + \frac{d}{d h} [- 2 a h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.3

Since $a^{2}$ is constant with respect to $h$ , the derivative of $a^{2}$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{5 (0 + \frac{d}{d h} [- 2 a h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.4

Since $- 2 a$ is constant with respect to $h$ , the derivative of $- 2 a h$ with respect to $h$ is $- 2 a \frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{5 (0 - 2 a \frac{d}{d h} [h] + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.5

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

$lim_{h \to 0} \frac{5 (0 - 2 a \cdot 1 + \frac{d}{d h} [h^{2}]) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.6

Differentiate using the Power Rule which states that $\frac{d}{d h} [h^{n}]$ is $n h^{n - 1}$ where $n = 2$ .

$lim_{h \to 0} \frac{5 (0 - 2 a \cdot 1 + 2 h) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.7

Multiply $- 2$ by $1$ .

$lim_{h \to 0} \frac{5 (0 - 2 a + 2 h) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6.8

Subtract $2 a$ from $0$ .

$lim_{h \to 0} \frac{5 (- 2 a + 2 h) + \frac{d}{d h} [- 5 a^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.7

Since $- 5 a^{2}$ is constant with respect to $h$ , the derivative of $- 5 a^{2}$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{5 (- 2 a + 2 h) + 0}{\frac{d}{d h} [h]}$

Step 1.3.8

Simplify.

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Step 1.3.8.1

Apply the distributive property.

$lim_{h \to 0} \frac{5 (- 2 a) + 5 (2 h) + 0}{\frac{d}{d h} [h]}$

Step 1.3.8.2

Combine terms.

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Step 1.3.8.2.1

Multiply $- 2$ by $5$ .

$lim_{h \to 0} \frac{- 10 a + 5 (2 h) + 0}{\frac{d}{d h} [h]}$

Step 1.3.8.2.2

Multiply $2$ by $5$ .

$lim_{h \to 0} \frac{- 10 a + 10 h + 0}{\frac{d}{d h} [h]}$

Step 1.3.8.2.3

Add $- 10 a + 10 h$ and $0$ .

$lim_{h \to 0} \frac{- 10 a + 10 h}{\frac{d}{d h} [h]}$

Step 1.3.9

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

$lim_{h \to 0} \frac{- 10 a + 10 h}{1}$

Step 1.4

Divide $- 10 a + 10 h$ by $1$ .

$lim_{h \to 0} - 10 a + 10 h$

Step 2

Evaluate the limit.

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Step 2.1

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

$- lim_{h \to 0} 10 a + lim_{h \to 0} 10 h$

Step 2.2

Evaluate the limit of $10 a$ which is constant as $h$ approaches $0$ .

$- 10 a + lim_{h \to 0} 10 h$

Step 2.3

Move the term $10$ outside of the limit because it is constant with respect to $h$ .

$- 10 a + 10 lim_{h \to 0} h$

Step 3

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

$- 10 a + 10 \cdot 0$

Step 4

Simplify the answer.

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Step 4.1

Multiply $10$ by $0$ .

$- 10 a + 0$

Step 4.2

Add $- 10 a$ and $0$ .

$- 10 a$