Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

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

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

Step 1.2

Combine $- 1$ and $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

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

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

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.

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

Move the limit under the radical sign.

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Any root of $1$ is $1$ .

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

Step 3.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.2.3.2

Subtract $1$ from $1$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

Move the limit under the radical sign.

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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Step 3.1.3.5.1

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

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

Step 3.1.3.5.2

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

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

Step 3.1.3.6

Simplify the answer.

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Step 3.1.3.6.1

Any root of $1$ is $1$ .

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

Step 3.1.3.6.2

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

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

Step 3.1.3.6.3

Multiply $- 1$ by $1$ .

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

Step 3.1.3.6.4

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 3.1.3.6.5

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

Undefined

$\frac{0}{0}$

Step 3.1.3.7

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 3.3.4.4

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

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

Step 3.3.4.5

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

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

Step 3.3.4.6

Combine the numerators over the common denominator.

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

Step 3.3.4.7

Simplify the numerator.

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Step 3.3.4.7.1

Multiply $- 1$ by $2$ .

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

Step 3.3.4.7.2

Subtract $2$ from $1$ .

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

Step 3.3.4.8

Move the negative in front of the fraction.

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

Step 3.3.4.9

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

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

Step 3.3.4.10

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

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 1$ .

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

Add $1$ and $0$ .

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

Step 3.3.12

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

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

Step 3.3.13

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

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

Step 3.3.14

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

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

Step 3.3.15

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

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

Step 3.3.16

Combine the numerators over the common denominator.

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

Step 3.3.17

Simplify the numerator.

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Step 3.3.17.1

Multiply $- 1$ by $2$ .

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

Step 3.3.17.2

Subtract $2$ from $1$ .

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

Step 3.3.18

Move the negative in front of the fraction.

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

Step 3.3.19

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

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

Step 3.3.20

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

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

Step 3.3.21

Simplify.

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Step 3.3.21.1

Apply the distributive property.

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

Step 3.3.21.2

Combine terms.

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Step 3.3.21.2.1

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

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

Step 3.3.21.2.2

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

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

Step 3.3.21.2.3

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

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Step 3.3.21.2.3.1

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

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Step 3.3.21.2.3.1.1

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

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

Step 3.3.21.2.3.1.2

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

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

Step 3.3.21.2.3.2

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

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

Step 3.3.21.2.3.3

Combine the numerators over the common denominator.

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

Step 3.3.21.2.3.4

Subtract $1$ from $2$ .

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

Step 3.3.21.2.4

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

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

Step 3.3.21.2.5

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

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

Step 3.3.21.2.6

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

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

Step 3.3.21.2.7

Combine the numerators over the common denominator.

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

Step 3.3.21.2.8

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

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

Step 3.3.21.2.9

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

$lim_{x \to 1} \frac{- \frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 x^{\frac{1}{2}}}{2} - \frac{1}{2 x^{\frac{1}{2}}}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Convert fractional exponents to radicals.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.6

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

$lim_{x \to 1} - \frac{1}{(\frac{3 \sqrt{x}}{2} - \frac{1}{2 \sqrt{x}}) (2 \sqrt{x})}$

Step 3.7

Combine terms.

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Step 3.7.1

To write $\frac{3 \sqrt{x}}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 3.7.2

Multiply $\frac{3 \sqrt{x}}{2}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$lim_{x \to 1} - \frac{1}{(\frac{3 \sqrt{x} \sqrt{x}}{2 \sqrt{x}} - \frac{1}{2 \sqrt{x}}) (2 \sqrt{x})}$

Step 3.7.3

Combine the numerators over the common denominator.

$lim_{x \to 1} - \frac{1}{\frac{3 \sqrt{x} \sqrt{x} - 1}{2 \sqrt{x}} (2 \sqrt{x})}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 1} \frac{1}{\frac{3 \sqrt{x} \sqrt{x} - 1}{2 \sqrt{x}} (2 \sqrt{x})}$

Step 4.2

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

$- \frac{1}{2} lim_{x \to 1} \frac{1}{\frac{3 \sqrt{x} \sqrt{x} - 1}{2 \sqrt{x}} (\sqrt{x})}$

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.10

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

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

Step 4.11

Move the limit under the radical sign.

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

Step 4.12

Move the limit under the radical sign.

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

Step 4.13

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

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

Step 4.14

Move the limit under the radical sign.

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

Step 4.15

Move the limit under the radical sign.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Any root of $1$ is $1$ .

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

Step 6.1.2

Multiply $3$ by $1$ .

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

Step 6.1.3

Any root of $1$ is $1$ .

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

Step 6.1.4

Multiply $3$ by $1$ .

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

Step 6.1.5

Multiply $- 1$ by $1$ .

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

Step 6.1.6

Subtract $1$ from $3$ .

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

Step 6.2

Any root of $1$ is $1$ .

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

Step 6.3

Simplify the denominator.

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.4

Any root of $1$ is $1$ .

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

Step 6.5

Multiply $2$ by $1$ .

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

Step 6.6

Divide $2$ by $2$ .

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

Step 6.7

Cancel the common factor of $1$ .

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Step 6.7.1

Cancel the common factor.

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

Step 6.7.2

Rewrite the expression.

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

Step 6.8

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

$- \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$