Calculus Examples

$lim_{x \to 1} \frac{x^{\frac{1}{2} \cdot 6} - x^{\frac{1}{2}}}{x^{\frac{1}{2} \cdot 3} - x^{\frac{1}{2}}}$

Step 1

Simplify the limit argument.

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Step 1.1

Convert fractional exponents to radicals.

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Step 1.1.1

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

$lim_{x \to 1} \frac{x^{\frac{1}{2} \cdot 6} - \sqrt{x}}{x^{\frac{1}{2} \cdot 3} - x^{\frac{1}{2}}}$

Step 1.1.2

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

$lim_{x \to 1} \frac{x^{\frac{1}{2} \cdot 6} - \sqrt{x}}{x^{\frac{1}{2} \cdot 3} - \sqrt{x}}$

Step 1.2

Combine factors.

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Step 1.2.1

Combine $\frac{1}{2}$ and $6$ .

$lim_{x \to 1} \frac{x^{\frac{6}{2}} - \sqrt{x}}{x^{\frac{1}{2} \cdot 3} - \sqrt{x}}$

Step 1.2.2

Combine $\frac{1}{2}$ and $3$ .

$lim_{x \to 1} \frac{x^{\frac{6}{2}} - \sqrt{x}}{x^{\frac{3}{2}} - \sqrt{x}}$

Step 1.3

Cancel the common factor of $6$ and $2$ .

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Step 1.3.1

Factor $2$ out of $6$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $2$ out of $2$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.3.2.4

Divide $3$ by $1$ .

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

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

Step 2.1.2.3

Move the limit under the radical sign.

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

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

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

Step 2.1.2.4.2

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

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

Step 2.1.2.5

Simplify the answer.

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Step 2.1.2.5.1

Simplify each term.

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Step 2.1.2.5.1.1

One to any power is one.

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

Step 2.1.2.5.1.2

Any root of $1$ is $1$ .

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

Step 2.1.2.5.1.3

Multiply $- 1$ by $1$ .

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

Step 2.1.2.5.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} x^{\frac{3}{2}} - \sqrt{x}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$\frac{0}{lim_{x \to 1} x^{\frac{3}{2}} - lim_{x \to 1} \sqrt{x}}$

Step 2.1.3.2

Move the exponent $\frac{3}{2}$ from $x^{\frac{3}{2}}$ outside the limit using the Limits Power Rule.

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

Step 2.1.3.3

Move the limit under the radical sign.

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

Step 2.1.3.4

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

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Step 2.1.3.4.1

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

$\frac{0}{1^{\frac{3}{2}} - \sqrt{lim_{x \to 1} x}}$

Step 2.1.3.4.2

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

$\frac{0}{1^{\frac{3}{2}} - \sqrt{1}}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Simplify each term.

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Step 2.1.3.5.1.1

One to any power is one.

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

Step 2.1.3.5.1.2

Any root of $1$ is $1$ .

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

Step 2.1.3.5.1.3

Multiply $- 1$ by $1$ .

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

Step 2.1.3.5.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 2.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

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

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

Step 2.3.4.4

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

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

Step 2.3.4.5

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

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

Step 2.3.4.6

Combine the numerators over the common denominator.

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

Step 2.3.4.7

Simplify the numerator.

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Step 2.3.4.7.1

Multiply $- 1$ by $2$ .

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

Step 2.3.4.7.2

Subtract $2$ from $1$ .

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

Step 2.3.4.8

Move the negative in front of the fraction.

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

Step 2.3.4.9

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

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

Step 2.3.4.10

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

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

Step 2.3.5

By the Sum Rule, the derivative of $x^{\frac{3}{2}} - \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [- \sqrt{x}]$ .

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

Step 2.3.6

Evaluate $\frac{d}{d x} [x^{\frac{3}{2}}]$ .

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Step 2.3.6.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 2.3.6.2

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

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

Step 2.3.6.3

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

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

Step 2.3.6.4

Combine the numerators over the common denominator.

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

Step 2.3.6.5

Simplify the numerator.

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Step 2.3.6.5.1

Multiply $- 1$ by $2$ .

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

Step 2.3.6.5.2

Subtract $2$ from $3$ .

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

Step 2.3.7

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

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Step 2.3.7.1

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

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

Step 2.3.7.2

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

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

Step 2.3.7.3

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

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

Step 2.3.7.4

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

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

Step 2.3.7.5

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

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

Step 2.3.7.6

Combine the numerators over the common denominator.

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

Step 2.3.7.7

Simplify the numerator.

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Step 2.3.7.7.1

Multiply $- 1$ by $2$ .

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

Step 2.3.7.7.2

Subtract $2$ from $1$ .

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

Step 2.3.7.8

Move the negative in front of the fraction.

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

Step 2.3.7.9

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

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

Step 2.3.7.10

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

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

Step 2.3.8

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

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

Step 2.4

Convert fractional exponents to radicals.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Combine terms.

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Step 2.5.1

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

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

Step 2.5.2

Combine $3 x^{2}$ and $\frac{2 \sqrt{x}}{2 \sqrt{x}}$ .

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

Step 2.5.3

Combine the numerators over the common denominator.

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

Combine the numerators over the common denominator.

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Move the term $3 \cdot 2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Move the limit under the radical sign.

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

Step 3.9

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

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

Step 3.10

Move the limit under the radical sign.

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

Move the limit under the radical sign.

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

Step 3.17

Move the limit under the radical sign.

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

Step 3.18

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

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

Step 3.19

Move the limit under the radical sign.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

Simplify the answer.

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Step 5.1

Cancel the common factor of $\frac{1}{2}$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3

Cancel the common factor of $\sqrt{1}$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

$(3 \cdot 2 \cdot 1^{2} \cdot \sqrt{1} - 1 \cdot 1) \frac{1}{3 \sqrt{1} \cdot \sqrt{1} - 1 \cdot 1}$

Step 5.4

Multiply $3$ by $2$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1 \cdot 1) \frac{1}{3 \sqrt{1} \cdot \sqrt{1} - 1 \cdot 1}$

Step 5.5

Multiply $- 1$ by $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 \sqrt{1} \cdot \sqrt{1} - 1 \cdot 1}$

Step 5.6

Raise $\sqrt{1}$ to the power of $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 ({\sqrt{1}}^{1} \sqrt{1}) - 1 \cdot 1}$

Step 5.7

Raise $\sqrt{1}$ to the power of $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 ({\sqrt{1}}^{1} {\sqrt{1}}^{1}) - 1 \cdot 1}$

Step 5.8

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

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 {\sqrt{1}}^{1 + 1} - 1 \cdot 1}$

Step 5.9

Add $1$ and $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 {\sqrt{1}}^{2} - 1 \cdot 1}$

Step 5.10

Multiply $- 1$ by $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 {\sqrt{1}}^{2} - 1}$

Step 5.11

Simplify the denominator.

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Step 5.11.1

Any root of $1$ is $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 \cdot 1^{2} - 1}$

Step 5.11.2

One to any power is one.

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 \cdot 1 - 1}$

Step 5.11.3

Multiply $3$ by $1$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{3 - 1}$

Step 5.11.4

Subtract $1$ from $3$ .

$(6 \cdot 1^{2} \cdot \sqrt{1} - 1) \frac{1}{2}$

Step 5.12

Simplify each term.

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Step 5.12.1

One to any power is one.

$(6 \cdot 1 \cdot \sqrt{1} - 1) \frac{1}{2}$

Step 5.12.2

Multiply $6$ by $1$ .

$(6 \cdot \sqrt{1} - 1) \frac{1}{2}$

Step 5.12.3

Any root of $1$ is $1$ .

$(6 \cdot 1 - 1) \frac{1}{2}$

Step 5.12.4

Multiply $6$ by $1$ .

$(6 - 1) \frac{1}{2}$

Step 5.13

Subtract $1$ from $6$ .

$5 (\frac{1}{2})$

Step 5.14

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$