Calculus Examples

$lim_{x \to 4} \frac{\frac{1}{\sqrt{4}} - \frac{1}{2}}{x - 4}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{\sqrt{4}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{1}{\sqrt{4}} \cdot \frac{2}{2} - \frac{1}{2}}{x - 4}$

Step 1.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{4}}{\sqrt{4}}$ .

$lim_{x \to 4} \frac{\frac{1}{\sqrt{4}} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{\sqrt{4}}{\sqrt{4}}}{x - 4}$

Step 1.3

Write each expression with a common denominator of $\sqrt{4} \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{\sqrt{4}}$ by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{2}{\sqrt{4} \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{4}}{\sqrt{4}}}{x - 4}$

Step 1.3.2

Multiply $\frac{1}{2}$ by $\frac{\sqrt{4}}{\sqrt{4}}$ .

$lim_{x \to 4} \frac{\frac{2}{\sqrt{4} \cdot 2} - \frac{\sqrt{4}}{2 \sqrt{4}}}{x - 4}$

Step 1.3.3

Reorder the factors of $\sqrt{4} \cdot 2$ .

$lim_{x \to 4} \frac{\frac{2}{2 \sqrt{4}} - \frac{\sqrt{4}}{2 \sqrt{4}}}{x - 4}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{\frac{2 - \sqrt{4}}{2 \sqrt{4}}}{x - 4}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 4} \frac{2 - \sqrt{4}}{2 \sqrt{4}} \cdot \frac{1}{x - 4}$

Step 2.2

Multiply $\frac{2 - \sqrt{4}}{2 \sqrt{4}}$ by $\frac{1}{x - 4}$ .

$lim_{x \to 4} \frac{2 - \sqrt{4}}{2 \sqrt{4} (x - 4)}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$\frac{lim_{x \to 4} 2 - \sqrt{4}}{lim_{x \to 4} 2 \sqrt{4} (x - 4)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit of $2 - \sqrt{4}$ which is constant as $x$ approaches $4$ .

$\frac{2 - \sqrt{4}}{lim_{x \to 4} 2 \sqrt{4} (x - 4)}$

Step 3.1.2.2

Simplify the answer.

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Step 3.1.2.2.1

Simplify each term.

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Step 3.1.2.2.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{2 - \sqrt{2^{2}}}{lim_{x \to 4} 2 \sqrt{4} (x - 4)}$

Step 3.1.2.2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 - 1 \cdot 2}{lim_{x \to 4} 2 \sqrt{4} (x - 4)}$

Step 3.1.2.2.1.3

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 4} 2 \sqrt{4} (x - 4)}$

Step 3.1.2.2.2

Subtract $2$ from $2$ .

$\frac{0}{lim_{x \to 4} 2 \sqrt{4} (x - 4)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

Move the term $2 \sqrt{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{2 \sqrt{4} lim_{x \to 4} x - 4}$

Step 3.1.3.1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{2 \sqrt{4} (lim_{x \to 4} x - lim_{x \to 4} 4)}$

Step 3.1.3.1.3

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

$\frac{0}{2 \sqrt{4} (lim_{x \to 4} x - 1 \cdot 4)}$

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{2 \sqrt{4} (4 - 1 \cdot 4)}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{0}{2 \sqrt{2^{2}} (4 - 1 \cdot 4)}$

Step 3.1.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{2 \cdot 2 (4 - 1 \cdot 4)}$

Step 3.1.3.3.3

Multiply $2$ by $2$ .

$\frac{0}{4 (4 - 1 \cdot 4)}$

Step 3.1.3.3.4

Multiply $- 1$ by $4$ .

$\frac{0}{4 (4 - 4)}$

Step 3.1.3.3.5

Subtract $4$ from $4$ .

$\frac{0}{4 \cdot 0}$

Step 3.1.3.3.6

Multiply $4$ by $0$ .

$\frac{0}{0}$

Step 3.1.3.3.7

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

Undefined

$\frac{0}{0}$

Step 3.1.3.4

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2

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.

$lim_{x \to 4} \frac{2 - \sqrt{4}}{2 \sqrt{4} (x - 4)} = lim_{x \to 4} \frac{\frac{d}{d x} [2 - \sqrt{4}]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [2 - \sqrt{4}]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.2

Rewrite $4$ as $2^{2}$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [2 - \sqrt{2^{2}}]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.3

Pull terms out from under the radical, assuming positive real numbers.

$lim_{x \to 4} \frac{\frac{d}{d x} [2 - 1 \cdot 2]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.4

By the Sum Rule, the derivative of $2 - 1 \cdot 2$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 1 \cdot 2]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [2] + \frac{d}{d x} [- 1 \cdot 2]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{0 + \frac{d}{d x} [- 1 \cdot 2]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.6

Evaluate $\frac{d}{d x} [- 1 \cdot 2]$ .

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Step 3.3.6.1

Multiply $- 1$ by $2$ .

$lim_{x \to 4} \frac{0 + \frac{d}{d x} [- 2]}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.6.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{0 + 0}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.7

Add $0$ and $0$ .

$lim_{x \to 4} \frac{0}{\frac{d}{d x} [2 \sqrt{4} (x - 4)]}$

Step 3.3.8

Rewrite $4$ as $2^{2}$ .

$lim_{x \to 4} \frac{0}{\frac{d}{d x} [2 \sqrt{2^{2}} (x - 4)]}$

Step 3.3.9

Pull terms out from under the radical, assuming positive real numbers.

$lim_{x \to 4} \frac{0}{\frac{d}{d x} [2 \cdot 2 (x - 4)]}$

Step 3.3.10

Multiply $2$ by $2$ .

$lim_{x \to 4} \frac{0}{\frac{d}{d x} [4 (x - 4)]}$

Step 3.3.11

Since $4$ is constant with respect to $x$ , the derivative of $4 (x - 4)$ with respect to $x$ is $4 \frac{d}{d x} [x - 4]$ .

$lim_{x \to 4} \frac{0}{4 \frac{d}{d x} [x - 4]}$

Step 3.3.12

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

$lim_{x \to 4} \frac{0}{4 (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])}$

Step 3.3.13

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

$lim_{x \to 4} \frac{0}{4 (1 + \frac{d}{d x} [- 4])}$

Step 3.3.14

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

$lim_{x \to 4} \frac{0}{4 (1 + 0)}$

Step 3.3.15

Add $1$ and $0$ .

$lim_{x \to 4} \frac{0}{4 \cdot 1}$

Step 3.3.16

Multiply $4$ by $1$ .

$lim_{x \to 4} \frac{0}{4}$

Step 3.4

Cancel the common factor of $0$ and $4$ .

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Step 3.4.1

Factor $4$ out of $0$ .

$lim_{x \to 4} \frac{4 (0)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $4$ out of $4$ .

$lim_{x \to 4} \frac{4 \cdot 0}{4 \cdot 1}$

Step 3.4.2.2

Cancel the common factor.

$lim_{x \to 4} \frac{4 \cdot 0}{4 \cdot 1}$

Step 3.4.2.3

Rewrite the expression.

$lim_{x \to 4} \frac{0}{1}$

Step 3.4.2.4

Divide $0$ by $1$ .

$lim_{x \to 4} 0$

Step 4

Evaluate the limit of $0$ which is constant as $x$ approaches $4$ .

$0$