Calculus Examples

$lim_{z \to 3} \frac{z - 3}{\sqrt{4 - z} - 1}$

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.

$\frac{lim_{z \to 3} z - 3}{lim_{z \to 3} \sqrt{4 - z} - 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $3$ .

$\frac{lim_{z \to 3} z - lim_{z \to 3} 3}{lim_{z \to 3} \sqrt{4 - z} - 1}$

Step 1.1.2.1.2

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

$\frac{lim_{z \to 3} z - 1 \cdot 3}{lim_{z \to 3} \sqrt{4 - z} - 1}$

Step 1.1.2.2

Evaluate the limit of $z$ by plugging in $3$ for $z$ .

$\frac{3 - 1 \cdot 3}{lim_{z \to 3} \sqrt{4 - z} - 1}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{z \to 3} \sqrt{4 - z} - 1}$

Step 1.1.2.3.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{z \to 3} \sqrt{4 - z} - 1}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $3$ .

$\frac{0}{lim_{z \to 3} \sqrt{4 - z} - lim_{z \to 3} 1}$

Step 1.1.3.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{z \to 3} 4 - z} - lim_{z \to 3} 1}$

Step 1.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $3$ .

$\frac{0}{\sqrt{lim_{z \to 3} 4 - lim_{z \to 3} z} - lim_{z \to 3} 1}$

Step 1.1.3.4

Evaluate the limit of $4$ which is constant as $z$ approaches $3$ .

$\frac{0}{\sqrt{4 - lim_{z \to 3} z} - lim_{z \to 3} 1}$

Step 1.1.3.5

Evaluate the limit of $1$ which is constant as $z$ approaches $3$ .

$\frac{0}{\sqrt{4 - lim_{z \to 3} z} - 1 \cdot 1}$

Step 1.1.3.6

Simplify terms.

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Step 1.1.3.6.1

Evaluate the limit of $z$ by plugging in $3$ for $z$ .

$\frac{0}{\sqrt{4 - 3} - 1 \cdot 1}$

Step 1.1.3.6.2

Simplify the answer.

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Step 1.1.3.6.2.1

Simplify each term.

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Step 1.1.3.6.2.1.1

Subtract $3$ from $4$ .

$\frac{0}{\sqrt{1} - 1 \cdot 1}$

Step 1.1.3.6.2.1.2

Any root of $1$ is $1$ .

$\frac{0}{1 - 1 \cdot 1}$

Step 1.1.3.6.2.1.3

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 1.1.3.6.2.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.6.2.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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

Undefined

$\frac{0}{0}$

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.

$lim_{z \to 3} \frac{z - 3}{\sqrt{4 - z} - 1} = lim_{z \to 3} \frac{\frac{d}{d z} [z - 3]}{\frac{d}{d z} [\sqrt{4 - z} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{z \to 3} \frac{\frac{d}{d z} [z - 3]}{\frac{d}{d z} [\sqrt{4 - z} - 1]}$

Step 1.3.2

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

$lim_{z \to 3} \frac{\frac{d}{d z} [z] + \frac{d}{d z} [- 3]}{\frac{d}{d z} [\sqrt{4 - z} - 1]}$

Step 1.3.3

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

$lim_{z \to 3} \frac{1 + \frac{d}{d z} [- 3]}{\frac{d}{d z} [\sqrt{4 - z} - 1]}$

Step 1.3.4

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

$lim_{z \to 3} \frac{1 + 0}{\frac{d}{d z} [\sqrt{4 - z} - 1]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{z \to 3} \frac{1}{\frac{d}{d z} [\sqrt{4 - z} - 1]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{4 - z} - 1$ with respect to $z$ is $\frac{d}{d z} [\sqrt{4 - z}] + \frac{d}{d z} [- 1]$ .

$lim_{z \to 3} \frac{1}{\frac{d}{d z} [\sqrt{4 - z}] + \frac{d}{d z} [- 1]}$

Step 1.3.7

Evaluate $\frac{d}{d z} [\sqrt{4 - z}]$ .

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Step 1.3.7.1

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

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

Step 1.3.7.2

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

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Step 1.3.7.2.1

To apply the Chain Rule, set $u$ as $4 - z$ .

$lim_{z \to 3} \frac{1}{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d z} [4 - z] + \frac{d}{d z} [- 1]}$

Step 1.3.7.2.2

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

$lim_{z \to 3} \frac{1}{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d z} [4 - z] + \frac{d}{d z} [- 1]}$

Step 1.3.7.2.3

Replace all occurrences of $u$ with $4 - z$ .

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

Step 1.3.7.3

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

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

Step 1.3.7.4

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

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

Step 1.3.7.5

Since $- 1$ is constant with respect to $z$ , the derivative of $- z$ with respect to $z$ is $- \frac{d}{d z} [z]$ .

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

Step 1.3.7.6

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

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

Step 1.3.7.7

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

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

Step 1.3.7.8

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

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

Step 1.3.7.9

Combine the numerators over the common denominator.

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

Step 1.3.7.10

Simplify the numerator.

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Step 1.3.7.10.1

Multiply $- 1$ by $2$ .

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

Step 1.3.7.10.2

Subtract $2$ from $1$ .

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

Step 1.3.7.11

Move the negative in front of the fraction.

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

Step 1.3.7.12

Multiply $- 1$ by $1$ .

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

Step 1.3.7.13

Subtract $1$ from $0$ .

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

Step 1.3.7.14

Combine $\frac{1}{2}$ and ${(4 - z)}^{- \frac{1}{2}}$ .

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

Step 1.3.7.15

Combine $\frac{{(4 - z)}^{- \frac{1}{2}}}{2}$ and $- 1$ .

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

Step 1.3.7.16

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

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

Step 1.3.7.17

Rewrite $- 1 {(4 - z)}^{- \frac{1}{2}}$ as $- {(4 - z)}^{- \frac{1}{2}}$ .

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

Step 1.3.7.18

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

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

Step 1.3.7.19

Move the negative in front of the fraction.

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

Step 1.3.8

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

$lim_{z \to 3} \frac{1}{- \frac{1}{2 {(4 - z)}^{\frac{1}{2}}} + 0}$

Step 1.3.9

Add $- \frac{1}{2 {(4 - z)}^{\frac{1}{2}}}$ and $0$ .

$lim_{z \to 3} \frac{1}{- \frac{1}{2 {(4 - z)}^{\frac{1}{2}}}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{z \to 3} 1 (- (2 {(4 - z)}^{\frac{1}{2}}))$

Step 1.5

Rewrite ${(4 - z)}^{\frac{1}{2}}$ as $\sqrt{4 - z}$ .

$lim_{z \to 3} 1 (- (2 \sqrt{4 - z}))$

Step 1.6

Combine factors.

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Step 1.6.1

Multiply $2$ by $- 1$ .

$lim_{z \to 3} 1 (- 2 \sqrt{4 - z})$

Step 1.6.2

Multiply $- 2$ by $1$ .

$lim_{z \to 3} - 2 \sqrt{4 - z}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $- 2$ outside of the limit because it is constant with respect to $z$ .

$- 2 lim_{z \to 3} \sqrt{4 - z}$

Step 2.2

Move the limit under the radical sign.

$- 2 \sqrt{lim_{z \to 3} 4 - z}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $3$ .

$- 2 \sqrt{lim_{z \to 3} 4 - lim_{z \to 3} z}$

Step 2.4

Evaluate the limit of $4$ which is constant as $z$ approaches $3$ .

$- 2 \sqrt{4 - lim_{z \to 3} z}$

Step 2.5

Simplify terms.

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Step 2.5.1

Evaluate the limit of $z$ by plugging in $3$ for $z$ .

$- 2 \sqrt{4 - 3}$

Step 2.5.2

Simplify the answer.

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Step 2.5.2.1

Subtract $3$ from $4$ .

$- 2 \sqrt{1}$

Step 2.5.2.2

Any root of $1$ is $1$ .

$- 2 \cdot 1$

Step 2.5.2.3

Multiply $- 2$ by $1$ .

$- 2$