Calculus Examples

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

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.

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

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 $1$ .

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

Step 1.1.2.2

Move the limit under the radical sign.

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

Step 1.1.2.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

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

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

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

Any root of $1$ is $1$ .

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

Step 1.1.2.5.1.2

One to any power is one.

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

Step 1.1.2.5.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2.5.2

Subtract $1$ from $1$ .

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

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 $1$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Move the limit under the radical sign.

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

Step 1.1.3.2

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

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

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

Any root of $1$ is $1$ .

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

Step 1.1.3.3.1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

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

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

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Combine the numerators over the common denominator.

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

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $2$ .

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

Step 1.3.3.6.2

Subtract $2$ from $1$ .

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

Step 1.3.3.7

Move the negative in front of the fraction.

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $2$ by $- 1$ .

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

Reorder terms.

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.8.3

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

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

Step 1.3.8.4

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

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

Step 1.3.8.5

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

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

Step 1.3.8.6

Combine the numerators over the common denominator.

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

Step 1.3.8.7

Simplify the numerator.

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Step 1.3.8.7.1

Multiply $- 1$ by $2$ .

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

Step 1.3.8.7.2

Subtract $2$ from $1$ .

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

Step 1.3.8.8

Move the negative in front of the fraction.

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

Step 1.3.8.9

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

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

Step 1.3.8.10

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

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

Step 1.3.9

Subtract $\frac{1}{2 x^{\frac{1}{2}}}$ from $0$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Convert fractional exponents to radicals.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.6

Multiply $2$ by $- 1$ .

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

Step 1.7

Combine terms.

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Step 1.7.1

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

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

Step 1.7.2

Combine $- 2 x$ and $\frac{2 \sqrt{x}}{2 \sqrt{x}}$ .

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

Step 1.7.3

Combine the numerators over the common denominator.

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

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 $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Move the limit under the radical sign.

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

Step 2.9

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

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

Step 2.10

Move the limit under the radical sign.

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

Step 2.11

Move the limit under the radical sign.

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

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 $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply $- 1$ by $2$ .

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

Step 4.2.1.2

Multiply $- 2$ by $2$ .

$- \frac{- 4 \cdot 1 \cdot \sqrt{1} + 1}{\sqrt{1}} \cdot \sqrt{1}$

Step 4.2.1.3

Multiply $- 4$ by $1$ .

$- \frac{- 4 \cdot \sqrt{1} + 1}{\sqrt{1}} \cdot \sqrt{1}$

Step 4.2.2

Any root of $1$ is $1$ .

$- \frac{- 4 \cdot 1 + 1}{\sqrt{1}} \cdot \sqrt{1}$

Step 4.2.3

Multiply $- 4$ by $1$ .

$- \frac{- 4 + 1}{\sqrt{1}} \cdot \sqrt{1}$

Step 4.2.4

Add $- 4$ and $1$ .

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

Step 4.3

Any root of $1$ is $1$ .

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

Step 4.4

Divide $- 3$ by $1$ .

$- - 3 \cdot \sqrt{1}$

Step 4.5

Multiply $- 1$ by $- 3$ .

$3 \cdot \sqrt{1}$

Step 4.6

Any root of $1$ is $1$ .

$3 \cdot 1$

Step 4.7

Multiply $3$ by $1$ .

$3$