Calculus Examples

$lim_{h \to 0} \frac{5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{4 - 3})}{h}$

Step 1

Subtract $3$ from $4$ .

$lim_{h \to 0} \frac{5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{1})}{h}$

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.

$\frac{lim_{h \to 0} 5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2

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

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Step 2.1.2.1

Evaluate the limit of $5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{1})$ by plugging in $0$ for $h$ .

$\frac{5 - \sqrt{4 - (3 + 0)} - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Add $3$ and $0$ .

$\frac{5 - \sqrt{4 - 1 \cdot 3} - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2.2.2

Multiply $- 1$ by $3$ .

$\frac{5 - \sqrt{4 - 3} - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2.2.3

Subtract $3$ from $4$ .

$\frac{5 - \sqrt{1} - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2.2.4

Any root of $1$ is $1$ .

$\frac{5 - 1 \cdot 1 - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2.2.5

Multiply $- 1$ by $1$ .

$\frac{5 - 1 - (5 - \sqrt{1})}{lim_{h \to 0} h}$

Step 2.1.2.2.6

Simplify each term.

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Step 2.1.2.2.6.1

Any root of $1$ is $1$ .

$\frac{5 - 1 - (5 - 1 \cdot 1)}{lim_{h \to 0} h}$

Step 2.1.2.2.6.2

Multiply $- 1$ by $1$ .

$\frac{5 - 1 - (5 - 1)}{lim_{h \to 0} h}$

Step 2.1.2.2.7

Subtract $1$ from $5$ .

$\frac{5 - 1 - 1 \cdot 4}{lim_{h \to 0} h}$

Step 2.1.2.2.8

Multiply $- 1$ by $4$ .

$\frac{5 - 1 - 4}{lim_{h \to 0} h}$

Step 2.1.2.3

Subtract $1$ from $5$ .

$\frac{4 - 4}{lim_{h \to 0} h}$

Step 2.1.2.4

Subtract $4$ from $4$ .

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

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.

$lim_{h \to 0} \frac{5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{1})}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{1})]}{\frac{d}{d h} [h]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 - \sqrt{4 - (3 + h)} - (5 - \sqrt{1})]}{\frac{d}{d h} [h]}$

Step 2.3.2

Simplify each term.

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Step 2.3.2.1

Any root of $1$ is $1$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 - \sqrt{4 - (3 + h)} - (5 - 1 \cdot 1)]}{\frac{d}{d h} [h]}$

Step 2.3.2.2

Multiply $- 1$ by $1$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 - \sqrt{4 - (3 + h)} - (5 - 1)]}{\frac{d}{d h} [h]}$

Step 2.3.3

Subtract $1$ from $5$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 - \sqrt{4 - (3 + h)} - 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.4

By the Sum Rule, the derivative of $5 - \sqrt{4 - (3 + h)} - 1 \cdot 4$ with respect to $h$ is $\frac{d}{d h} [5] + \frac{d}{d h} [- \sqrt{4 - (3 + h)}] + \frac{d}{d h} [- 1 \cdot 4]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5] + \frac{d}{d h} [- \sqrt{4 - (3 + h)}] + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.5

Since $5$ is constant with respect to $h$ , the derivative of $5$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{0 + \frac{d}{d h} [- \sqrt{4 - (3 + h)}] + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6

Evaluate $\frac{d}{d h} [- \sqrt{4 - (3 + h)}]$ .

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Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - (3 + h)}$ as ${(4 - (3 + h))}^{\frac{1}{2}}$ .

$lim_{h \to 0} \frac{0 + \frac{d}{d h} [- {(4 - (3 + h))}^{\frac{1}{2}}] + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.2

Since $- 1$ is constant with respect to $h$ , the derivative of $- {(4 - (3 + h))}^{\frac{1}{2}}$ with respect to $h$ is $- \frac{d}{d h} [{(4 - (3 + h))}^{\frac{1}{2}}]$ .

$lim_{h \to 0} \frac{0 - \frac{d}{d h} [{(4 - (3 + h))}^{\frac{1}{2}}] + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.3

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{\frac{1}{2}}$ and $g (h) = 4 - (3 + h)$ .

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Step 2.3.6.3.1

To apply the Chain Rule, set $u$ as $4 - (3 + h)$ .

$lim_{h \to 0} \frac{0 - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d h} [4 - (3 + h)]) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.3.2

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

$lim_{h \to 0} \frac{0 - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d h} [4 - (3 + h)]) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.3.3

Replace all occurrences of $u$ with $4 - (3 + h)$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} \frac{d}{d h} [4 - (3 + h)]) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.4

By the Sum Rule, the derivative of $4 - (3 + h)$ with respect to $h$ is $\frac{d}{d h} [4] + \frac{d}{d h} [- (3 + h)]$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} (\frac{d}{d h} [4] + \frac{d}{d h} [- (3 + h)])) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.5

Since $4$ is constant with respect to $h$ , the derivative of $4$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} (0 + \frac{d}{d h} [- (3 + h)])) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.6

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

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} (0 - \frac{d}{d h} [3 + h])) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.7

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

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} (0 - (\frac{d}{d h} [3] + \frac{d}{d h} [h]))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.8

Since $3$ is constant with respect to $h$ , the derivative of $3$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} (0 - (0 + \frac{d}{d h} [h]))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.9

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

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.11

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.12

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1 - 1 \cdot 2}{2}} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.13

Simplify the numerator.

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Step 2.3.6.13.1

Multiply $- 1$ by $2$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{1 - 2}{2}} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.13.2

Subtract $2$ from $1$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{\frac{- 1}{2}} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.14

Move the negative in front of the fraction.

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{- \frac{1}{2}} (0 - (0 + 1))) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.15

Add $0$ and $1$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{- \frac{1}{2}} (0 - 1 \cdot 1)) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.16

Multiply $- 1$ by $1$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{- \frac{1}{2}} (0 - 1)) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.17

Subtract $1$ from $0$ .

$lim_{h \to 0} \frac{0 - (\frac{1}{2} {(4 - (3 + h))}^{- \frac{1}{2}} \cdot - 1) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.18

Combine $\frac{1}{2}$ and ${(4 - (3 + h))}^{- \frac{1}{2}}$ .

$lim_{h \to 0} \frac{0 - (\frac{{(4 - (3 + h))}^{- \frac{1}{2}}}{2} \cdot - 1) + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.19

Combine $\frac{{(4 - (3 + h))}^{- \frac{1}{2}}}{2}$ and $- 1$ .

$lim_{h \to 0} \frac{0 - \frac{{(4 - (3 + h))}^{- \frac{1}{2}} \cdot - 1}{2} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.20

Move $- 1$ to the left of ${(4 - (3 + h))}^{- \frac{1}{2}}$ .

$lim_{h \to 0} \frac{0 - \frac{- 1 \cdot {(4 - (3 + h))}^{- \frac{1}{2}}}{2} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.21

Rewrite $- 1 {(4 - (3 + h))}^{- \frac{1}{2}}$ as $- {(4 - (3 + h))}^{- \frac{1}{2}}$ .

$lim_{h \to 0} \frac{0 - \frac{- {(4 - (3 + h))}^{- \frac{1}{2}}}{2} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.22

Move ${(4 - (3 + h))}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{h \to 0} \frac{0 - \frac{- 1}{2 {(4 - (3 + h))}^{\frac{1}{2}}} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.23

Move the negative in front of the fraction.

$lim_{h \to 0} \frac{0 - - \frac{1}{2 {(4 - (3 + h))}^{\frac{1}{2}}} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.24

Multiply $- 1$ by $- 1$ .

$lim_{h \to 0} \frac{0 + 1 \frac{1}{2 {(4 - (3 + h))}^{\frac{1}{2}}} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.6.25

Multiply $\frac{1}{2 {(4 - (3 + h))}^{\frac{1}{2}}}$ by $1$ .

$lim_{h \to 0} \frac{0 + \frac{1}{2 {(4 - (3 + h))}^{\frac{1}{2}}} + \frac{d}{d h} [- 1 \cdot 4]}{\frac{d}{d h} [h]}$

Step 2.3.7

Evaluate $\frac{d}{d h} [- 1 \cdot 4]$ .

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Step 2.3.7.1

Multiply $- 1$ by $4$ .

$lim_{h \to 0} \frac{0 + \frac{1}{2 {(4 - (3 + h))}^{\frac{1}{2}}} + \frac{d}{d h} [- 4]}{\frac{d}{d h} [h]}$

Step 2.3.7.2

Since $- 4$ is constant with respect to $h$ , the derivative of $- 4$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{0 + \frac{1}{2 {(4 - (3 + h))}^{\frac{1}{2}}} + 0}{\frac{d}{d h} [h]}$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Apply the distributive property.

$lim_{h \to 0} \frac{0 + \frac{1}{2 {(4 - 1 \cdot 3 - h)}^{\frac{1}{2}}} + 0}{\frac{d}{d h} [h]}$

Step 2.3.8.2

Combine terms.

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Step 2.3.8.2.1

Multiply $- 1$ by $3$ .

$lim_{h \to 0} \frac{0 + \frac{1}{2 {(4 - 3 - h)}^{\frac{1}{2}}} + 0}{\frac{d}{d h} [h]}$

Step 2.3.8.2.2

Subtract $3$ from $4$ .

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

Step 2.3.8.2.3

Add $0$ and $\frac{1}{2 {(1 - h)}^{\frac{1}{2}}}$ .

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

Step 2.3.8.2.4

Add $\frac{1}{2 {(1 - h)}^{\frac{1}{2}}}$ and $0$ .

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

Step 2.3.9

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

$lim_{h \to 0} \frac{\frac{1}{2 {(1 - h)}^{\frac{1}{2}}}}{1}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{1}{2 {(1 - h)}^{\frac{1}{2}}} \cdot 1$

Step 2.5

Rewrite ${(1 - h)}^{\frac{1}{2}}$ as $\sqrt{1 - h}$ .

$lim_{h \to 0} \frac{1}{2 \sqrt{1 - h}} \cdot 1$

Step 2.6

Multiply $\frac{1}{2 \sqrt{1 - h}}$ by $1$ .

$lim_{h \to 0} \frac{1}{2 \sqrt{1 - h}}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{2} lim_{h \to 0} \frac{1}{\sqrt{1 - h}}$

Step 3.2

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

$\frac{1}{2} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} \sqrt{1 - h}}$

Step 3.3

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

$\frac{1}{2} \cdot \frac{1}{lim_{h \to 0} \sqrt{1 - h}}$

Step 3.4

Move the limit under the radical sign.

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{h \to 0} 1 - h}}$

Step 3.5

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{h \to 0} 1 - lim_{h \to 0} h}}$

Step 3.6

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{1 - lim_{h \to 0} h}}$

Step 3.7

Simplify terms.

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Step 3.7.1

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

$\frac{1}{2} \cdot \frac{1}{\sqrt{1 + 0}}$

Step 3.7.2

Simplify the answer.

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Step 3.7.2.1

Simplify the denominator.

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Step 3.7.2.1.1

Add $1$ and $0$ .

$\frac{1}{2} \cdot \frac{1}{\sqrt{1}}$

Step 3.7.2.1.2

Any root of $1$ is $1$ .

$\frac{1}{2} \cdot \frac{1}{1}$

Step 3.7.2.2

Cancel the common factor of $1$ .

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Step 3.7.2.2.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{1}{1}$

Step 3.7.2.2.2

Rewrite the expression.

$\frac{1}{2} \cdot 1$

Step 3.7.2.3

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$