Calculus Examples

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

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.

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

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 $x$ approaches $1$ .

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

Step 1.1.2.2

Move the limit under the radical sign.

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

Step 1.1.2.3

Move the limit under the radical sign.

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

Step 1.1.2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 1.1.2.7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 1.1.2.7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Simplify each term.

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Step 1.1.2.8.1.1

Any root of $1$ is $1$ .

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

Step 1.1.2.8.1.2

Multiply $2$ by $1$ .

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

Step 1.1.2.8.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2.8.1.4

Subtract $1$ from $2$ .

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

Step 1.1.2.8.1.5

Any root of $1$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 1} x - 1}$

Step 1.1.2.8.1.6

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 1} x - 1}$

Step 1.1.2.8.2

Subtract $1$ from $1$ .

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

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 $1$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

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.

$lim_{x \to 1} \frac{\sqrt{x} - \sqrt{2 x - 1}}{x - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [\sqrt{x} - \sqrt{2 x - 1}]}{\frac{d}{d x} [x - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [\sqrt{x} - \sqrt{2 x - 1}]}{\frac{d}{d x} [x - 1]}$

Step 1.3.2

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

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

Step 1.3.3

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

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Combine the numerators over the common denominator.

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

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $2$ .

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

Step 1.3.3.6.2

Subtract $2$ from $1$ .

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

Step 1.3.3.7

Move the negative in front of the fraction.

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

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 $g (x) = 2 x - 1$ .

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Step 1.3.4.3.1

To apply the Chain Rule, set $u$ as $2 x - 1$ .

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

Step 1.3.4.3.2

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

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

Step 1.3.4.3.3

Replace all occurrences of $u$ with $2 x - 1$ .

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

Step 1.3.4.4

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

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

Step 1.3.4.5

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

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

Step 1.3.4.6

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

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

Step 1.3.4.7

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

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

Step 1.3.4.8

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

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

Step 1.3.4.9

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

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

Step 1.3.4.10

Combine the numerators over the common denominator.

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

Step 1.3.4.11

Simplify the numerator.

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Step 1.3.4.11.1

Multiply $- 1$ by $2$ .

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

Step 1.3.4.11.2

Subtract $2$ from $1$ .

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

Step 1.3.4.12

Move the negative in front of the fraction.

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

Step 1.3.4.13

Multiply $2$ by $1$ .

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

Step 1.3.4.14

Add $2$ and $0$ .

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

Step 1.3.4.15

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

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

Step 1.3.4.16

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

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

Step 1.3.4.17

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

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

Step 1.3.4.18

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

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

Step 1.3.4.19

Cancel the common factor.

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

Step 1.3.4.20

Rewrite the expression.

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3.5.2

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

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

Add $1$ and $0$ .

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

Step 1.4

Convert fractional exponents to radicals.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.5

Combine terms.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Combine using the product rule for radicals.

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

Step 1.5.3.3

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

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

Step 1.5.3.4

Combine using the product rule for radicals.

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

Step 1.5.4

Combine the numerators over the common denominator.

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

Step 1.6

Divide $\frac{\sqrt{2 x - 1} - 2 \sqrt{x}}{2 \sqrt{(2 x - 1) x}}$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

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

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 2.4

Move the limit under the radical sign.

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

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Move the limit under the radical sign.

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

Step 2.10

Move the limit under the radical sign.

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

Step 2.11

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 2.12

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 2.13

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

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

Step 2.14

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

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

Step 3

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3.4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $2$ by $1$ .

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

Step 4.1.2

Multiply $- 1$ by $1$ .

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

Step 4.1.3

Subtract $1$ from $2$ .

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

Step 4.1.4

Any root of $1$ is $1$ .

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

Step 4.1.5

Any root of $1$ is $1$ .

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

Step 4.1.6

Multiply $- 2$ by $1$ .

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

Step 4.1.7

Subtract $2$ from $1$ .

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2 \cdot 1 - 1 \cdot 1$ by $1$ .

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

Step 4.2.2

Multiply $2$ by $1$ .

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

Step 4.2.3

Multiply $- 1$ by $1$ .

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

Step 4.2.4

Subtract $1$ from $2$ .

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

Step 4.2.5

Any root of $1$ is $1$ .

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

Step 4.3

Divide $- 1$ by $1$ .

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

Step 4.4

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

$\frac{- 1}{2}$

Step 4.5

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$