Calculus Examples

$lim_{x \to 2} \frac{\sqrt{2 x} - 2}{\sqrt{14 x - 3} - 5}$

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.

$\frac{lim_{x \to 2} \sqrt{2 x} - 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

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 $x$ approaches $2$ .

$\frac{lim_{x \to 2} \sqrt{2 x} - lim_{x \to 2} 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2} 2 x} - lim_{x \to 2} 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.1.3

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

$\frac{\sqrt{2 lim_{x \to 2} x} - lim_{x \to 2} 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{2 lim_{x \to 2} x} - 1 \cdot 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{\sqrt{2 \cdot 2} - 1 \cdot 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply $2$ by $2$ .

$\frac{\sqrt{4} - 1 \cdot 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.3.1.2

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

$\frac{\sqrt{2^{2}} - 1 \cdot 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.3.1.3

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

$\frac{2 - 1 \cdot 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.3.1.4

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.2.3.2

Subtract $2$ from $2$ .

$\frac{0}{lim_{x \to 2} \sqrt{14 x - 3} - 5}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{0}{lim_{x \to 2} \sqrt{14 x - 3} - lim_{x \to 2} 5}$

Step 1.1.3.1.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 2} 14 x - 3} - lim_{x \to 2} 5}$

Step 1.1.3.1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{0}{\sqrt{lim_{x \to 2} 14 x - lim_{x \to 2} 3} - lim_{x \to 2} 5}$

Step 1.1.3.1.4

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\sqrt{14 lim_{x \to 2} x - lim_{x \to 2} 3} - lim_{x \to 2} 5}$

Step 1.1.3.1.5

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$\frac{0}{\sqrt{14 lim_{x \to 2} x - 1 \cdot 3} - lim_{x \to 2} 5}$

Step 1.1.3.1.6

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\frac{0}{\sqrt{14 lim_{x \to 2} x - 1 \cdot 3} - 1 \cdot 5}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{0}{\sqrt{14 \cdot 2 - 1 \cdot 3} - 1 \cdot 5}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $14$ by $2$ .

$\frac{0}{\sqrt{28 - 1 \cdot 3} - 1 \cdot 5}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $3$ .

$\frac{0}{\sqrt{28 - 3} - 1 \cdot 5}$

Step 1.1.3.3.1.3

Subtract $3$ from $28$ .

$\frac{0}{\sqrt{25} - 1 \cdot 5}$

Step 1.1.3.3.1.4

Rewrite $25$ as $5^{2}$ .

$\frac{0}{\sqrt{5^{2}} - 1 \cdot 5}$

Step 1.1.3.3.1.5

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

$\frac{0}{5 - 1 \cdot 5}$

Step 1.1.3.3.1.6

Multiply $- 1$ by $5$ .

$\frac{0}{5 - 5}$

Step 1.1.3.3.2

Subtract $5$ from $5$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 2} \frac{\sqrt{2 x} - 2}{\sqrt{14 x - 3} - 5} = lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{2 x} - 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{2 x} - 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.2

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

$lim_{x \to 2} \frac{\frac{d}{d x} [\sqrt{2 x}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt{2 x}]$ .

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Step 1.3.3.1

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

$lim_{x \to 2} \frac{\frac{d}{d x} [{(2 x)}^{\frac{1}{2}}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.2

Factor $2$ out of $2 x$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [{(2 (x))}^{\frac{1}{2}}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.3

Apply the product rule to $2 (x)$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [2^{\frac{1}{2}} x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.4

Since $2^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $2^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$lim_{x \to 2} \frac{2^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.5

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

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.6

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

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.7

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

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.8

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.9

Simplify the numerator.

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Step 1.3.3.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.9.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.10

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{2^{\frac{1}{2}} (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.11

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

$lim_{x \to 2} \frac{2^{\frac{1}{2}} \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.12

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

$lim_{x \to 2} \frac{\frac{2^{\frac{1}{2}} x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.13

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

$lim_{x \to 2} \frac{\frac{x^{- \frac{1}{2}}}{2 \cdot 2^{- \frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.14

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

$lim_{x \to 2} \frac{\frac{1}{2 \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.15

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.3.3.15.1

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

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Step 1.3.3.15.1.1

Raise $2$ to the power of $1$ .

$lim_{x \to 2} \frac{\frac{1}{2^{1} \cdot 2^{- \frac{1}{2}} x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.15.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 2} \frac{\frac{1}{2^{1 - \frac{1}{2}} x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.15.2

Write $1$ as a fraction with a common denominator.

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{2}{2} - \frac{1}{2}} x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.15.3

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{2 - 1}{2}} x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.3.15.4

Subtract $1$ from $2$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.4

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

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.5

Add $\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{14 x - 3} - 5]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{14 x - 3} - 5$ with respect to $x$ is $\frac{d}{d x} [\sqrt{14 x - 3}] + \frac{d}{d x} [- 5]$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{d}{d x} [\sqrt{14 x - 3}] + \frac{d}{d x} [- 5]}$

Step 1.3.7

Evaluate $\frac{d}{d x} [\sqrt{14 x - 3}]$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14 x - 3}$ as ${(14 x - 3)}^{\frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{d}{d x} [{(14 x - 3)}^{\frac{1}{2}}] + \frac{d}{d x} [- 5]}$

Step 1.3.7.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 14 x - 3$ .

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Step 1.3.7.2.1

To apply the Chain Rule, set $u$ as $14 x - 3$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [14 x - 3] + \frac{d}{d x} [- 5]}$

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 $n = \frac{1}{2}$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [14 x - 3] + \frac{d}{d x} [- 5]}$

Step 1.3.7.2.3

Replace all occurrences of $u$ with $14 x - 3$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} - 1} \frac{d}{d x} [14 x - 3] + \frac{d}{d x} [- 5]}$

Step 1.3.7.3

By the Sum Rule, the derivative of $14 x - 3$ with respect to $x$ is $\frac{d}{d x} [14 x] + \frac{d}{d x} [- 3]$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} - 1} (\frac{d}{d x} [14 x] + \frac{d}{d x} [- 3]) + \frac{d}{d x} [- 5]}$

Step 1.3.7.4

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} - 1} (14 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) + \frac{d}{d x} [- 5]}$

Step 1.3.7.5

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

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} - 1} (14 \cdot 1 + \frac{d}{d x} [- 3]) + \frac{d}{d x} [- 5]}$

Step 1.3.7.6

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

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} - 1} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.7

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

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.8

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

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.9

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1 - 1 \cdot 2}{2}} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.10

Simplify the numerator.

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Step 1.3.7.10.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{1 - 2}{2}} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.10.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{\frac{- 1}{2}} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.11

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{- \frac{1}{2}} (14 \cdot 1 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.12

Multiply $14$ by $1$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{- \frac{1}{2}} (14 + 0) + \frac{d}{d x} [- 5]}$

Step 1.3.7.13

Add $14$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{1}{2} {(14 x - 3)}^{- \frac{1}{2}} \cdot 14 + \frac{d}{d x} [- 5]}$

Step 1.3.7.14

Combine $\frac{1}{2}$ and ${(14 x - 3)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{{(14 x - 3)}^{- \frac{1}{2}}}{2} \cdot 14 + \frac{d}{d x} [- 5]}$

Step 1.3.7.15

Combine $\frac{{(14 x - 3)}^{- \frac{1}{2}}}{2}$ and $14$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{{(14 x - 3)}^{- \frac{1}{2}} \cdot 14}{2} + \frac{d}{d x} [- 5]}$

Step 1.3.7.16

Move $14$ to the left of ${(14 x - 3)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{14 \cdot {(14 x - 3)}^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 5]}$

Step 1.3.7.17

Move ${(14 x - 3)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{14}{2 {(14 x - 3)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}$

Step 1.3.7.18

Factor $2$ out of $14$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{2 \cdot 7}{2 {(14 x - 3)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}$

Step 1.3.7.19

Cancel the common factors.

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Step 1.3.7.19.1

Factor $2$ out of $2 {(14 x - 3)}^{\frac{1}{2}}$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{2 \cdot 7}{2 ({(14 x - 3)}^{\frac{1}{2}})} + \frac{d}{d x} [- 5]}$

Step 1.3.7.19.2

Cancel the common factor.

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{2 \cdot 7}{2 {(14 x - 3)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}$

Step 1.3.7.19.3

Rewrite the expression.

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{7}{{(14 x - 3)}^{\frac{1}{2}}} + \frac{d}{d x} [- 5]}$

Step 1.3.8

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

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{7}{{(14 x - 3)}^{\frac{1}{2}}} + 0}$

Step 1.3.9

Add $\frac{7}{{(14 x - 3)}^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 2} \frac{\frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}}}{\frac{7}{{(14 x - 3)}^{\frac{1}{2}}}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 2} \frac{1}{2^{\frac{1}{2}} x^{\frac{1}{2}}} \cdot \frac{{(14 x - 3)}^{\frac{1}{2}}}{7}$

Step 1.5

Convert fractional exponents to radicals.

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Step 1.5.1

Rewrite $2^{\frac{1}{2}}$ as $\sqrt{2}$ .

$lim_{x \to 2} \frac{1}{\sqrt{2} x^{\frac{1}{2}}} \cdot \frac{{(14 x - 3)}^{\frac{1}{2}}}{7}$

Step 1.5.2

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 2} \frac{1}{\sqrt{2} \sqrt{x}} \cdot \frac{{(14 x - 3)}^{\frac{1}{2}}}{7}$

Step 1.5.3

Rewrite ${(14 x - 3)}^{\frac{1}{2}}$ as $\sqrt{14 x - 3}$ .

$lim_{x \to 2} \frac{1}{\sqrt{2} \sqrt{x}} \cdot \frac{\sqrt{14 x - 3}}{7}$

Step 1.6

Combine factors.

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Step 1.6.1

Combine using the product rule for radicals.

$lim_{x \to 2} \frac{1}{\sqrt{2 x}} \cdot \frac{\sqrt{14 x - 3}}{7}$

Step 1.6.2

Multiply $\frac{1}{\sqrt{2 x}}$ by $\frac{\sqrt{14 x - 3}}{7}$ .

$lim_{x \to 2} \frac{\sqrt{14 x - 3}}{\sqrt{2 x} \cdot 7}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{7}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} lim_{x \to 2} \frac{\sqrt{14 x - 3}}{\sqrt{2 x}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{1}{7} \cdot \frac{lim_{x \to 2} \sqrt{14 x - 3}}{lim_{x \to 2} \sqrt{2 x}}$

Step 2.3

Move the limit under the radical sign.

$\frac{1}{7} \cdot \frac{\sqrt{lim_{x \to 2} 14 x - 3}}{lim_{x \to 2} \sqrt{2 x}}$

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{7} \cdot \frac{\sqrt{lim_{x \to 2} 14 x - lim_{x \to 2} 3}}{lim_{x \to 2} \sqrt{2 x}}$

Step 2.5

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{7} \cdot \frac{\sqrt{14 lim_{x \to 2} x - lim_{x \to 2} 3}}{lim_{x \to 2} \sqrt{2 x}}$

Step 2.6

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$\frac{1}{7} \cdot \frac{\sqrt{14 lim_{x \to 2} x - 1 \cdot 3}}{lim_{x \to 2} \sqrt{2 x}}$

Step 2.7

Move the limit under the radical sign.

$\frac{1}{7} \cdot \frac{\sqrt{14 lim_{x \to 2} x - 1 \cdot 3}}{\sqrt{lim_{x \to 2} 2 x}}$

Step 2.8

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

$\frac{1}{7} \cdot \frac{\sqrt{14 lim_{x \to 2} x - 1 \cdot 3}}{\sqrt{2 lim_{x \to 2} x}}$

Step 3

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{7} \cdot \frac{\sqrt{14 \cdot 2 - 1 \cdot 3}}{\sqrt{2 lim_{x \to 2} x}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{7} \cdot \frac{\sqrt{14 \cdot 2 - 1 \cdot 3}}{\sqrt{2 \cdot 2}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $14$ by $2$ .

$\frac{1}{7} \cdot \frac{\sqrt{28 - 1 \cdot 3}}{\sqrt{2 \cdot 2}}$

Step 4.1.2

Multiply $- 1$ by $3$ .

$\frac{1}{7} \cdot \frac{\sqrt{28 - 3}}{\sqrt{2 \cdot 2}}$

Step 4.1.3

Subtract $3$ from $28$ .

$\frac{1}{7} \cdot \frac{\sqrt{25}}{\sqrt{2 \cdot 2}}$

Step 4.1.4

Rewrite $25$ as $5^{2}$ .

$\frac{1}{7} \cdot \frac{\sqrt{5^{2}}}{\sqrt{2 \cdot 2}}$

Step 4.1.5

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

$\frac{1}{7} \cdot \frac{5}{\sqrt{2 \cdot 2}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $2$ .

$\frac{1}{7} \cdot \frac{5}{\sqrt{4}}$

Step 4.2.2

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

$\frac{1}{7} \cdot \frac{5}{\sqrt{2^{2}}}$

Step 4.2.3

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

$\frac{1}{7} \cdot \frac{5}{2}$

Step 4.3

Multiply $\frac{1}{7} \cdot \frac{5}{2}$ .

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Step 4.3.1

Multiply $\frac{1}{7}$ by $\frac{5}{2}$ .

$\frac{5}{7 \cdot 2}$

Step 4.3.2

Multiply $7$ by $2$ .

$\frac{5}{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{14}$

Decimal Form:

$0.3\overline{571428}$