Calculus Examples

$\frac{lim_{x \to 9} \sqrt[6]{7 x + 1} - 2}{\frac{1}{x} - \frac{1}{9}}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{lim_{x \to 9} \sqrt[6]{7 x + 1} - lim_{x \to 9} 2}{\frac{1}{x} - \frac{1}{9}}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt[6]{lim_{x \to 9} 7 x + 1} - lim_{x \to 9} 2}{\frac{1}{x} - \frac{1}{9}}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{\sqrt[6]{lim_{x \to 9} 7 x + lim_{x \to 9} 1} - lim_{x \to 9} 2}{\frac{1}{x} - \frac{1}{9}}$

Step 1.4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt[6]{7 lim_{x \to 9} x + lim_{x \to 9} 1} - lim_{x \to 9} 2}{\frac{1}{x} - \frac{1}{9}}$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $9$ .

$\frac{\sqrt[6]{7 lim_{x \to 9} x + 1} - lim_{x \to 9} 2}{\frac{1}{x} - \frac{1}{9}}$

Step 1.6

Evaluate the limit of $2$ which is constant as $x$ approaches $9$ .

$\frac{\sqrt[6]{7 lim_{x \to 9} x + 1} - 1 \cdot 2}{\frac{1}{x} - \frac{1}{9}}$

Step 2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{\sqrt[6]{7 \cdot 9 + 1} - 1 \cdot 2}{\frac{1}{x} - \frac{1}{9}}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $x \cdot 9$ .

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Step 3.1.1

Multiply $\frac{\sqrt[6]{7 \cdot 9 + 1} - 1 \cdot 2}{\frac{1}{x} - \frac{1}{9}}$ by $\frac{x \cdot 9}{x \cdot 9}$ .

$\frac{x \cdot 9}{x \cdot 9} \cdot \frac{\sqrt[6]{7 \cdot 9 + 1} - 1 \cdot 2}{\frac{1}{x} - \frac{1}{9}}$

Step 3.1.2

Combine.

$\frac{x \cdot 9 (\sqrt[6]{7 \cdot 9 + 1} - 1 \cdot 2)}{x \cdot 9 (\frac{1}{x} - \frac{1}{9})}$

Step 3.2

Apply the distributive property.

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{x \cdot 9 \frac{1}{x} + x \cdot 9 (- \frac{1}{9})}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Factor $x$ out of $x \cdot 9$ .

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{x (9) \frac{1}{x} + x \cdot 9 (- \frac{1}{9})}$

Step 3.3.1.2

Cancel the common factor.

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{x \cdot 9 \frac{1}{x} + x \cdot 9 (- \frac{1}{9})}$

Step 3.3.1.3

Rewrite the expression.

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{9 + x \cdot 9 (- \frac{1}{9})}$

Step 3.3.2

Cancel the common factor of $9$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{9 + x \cdot 9 \frac{- 1}{9}}$

Step 3.3.2.2

Factor $9$ out of $x \cdot 9$ .

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{9 + 9 \cdot x \frac{- 1}{9}}$

Step 3.3.2.3

Cancel the common factor.

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{9 + 9 \cdot x \frac{- 1}{9}}$

Step 3.3.2.4

Rewrite the expression.

$\frac{x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)}{9 + x \cdot - 1}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $x \cdot 9$ out of $x \cdot 9 \sqrt[6]{7 \cdot 9 + 1} + x \cdot 9 (- 1 \cdot 2)$ .

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Step 3.4.1.1

Factor $x \cdot 9$ out of $x \cdot 9 \sqrt[6]{7 \cdot 9 + 1}$ .

$\frac{x \cdot 9 (\sqrt[6]{7 \cdot 9 + 1}) + x \cdot 9 (- 1 \cdot 2)}{9 + x \cdot - 1}$

Step 3.4.1.2

Factor $x \cdot 9$ out of $x \cdot 9 (\sqrt[6]{7 \cdot 9 + 1}) + x \cdot 9 (- 1 \cdot 2)$ .

$\frac{x \cdot 9 (\sqrt[6]{7 \cdot 9 + 1} - 1 \cdot 2)}{9 + x \cdot - 1}$

Step 3.4.2

Combine exponents.

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Step 3.4.2.1

Multiply $7$ by $9$ .

$\frac{x \cdot 9 (\sqrt[6]{63 + 1} - 1 \cdot 2)}{9 + x \cdot - 1}$

Step 3.4.2.2

Multiply $- 1$ by $2$ .

$\frac{x \cdot 9 (\sqrt[6]{63 + 1} - 2)}{9 + x \cdot - 1}$

Step 3.4.3

Simplify each term.

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Step 3.4.3.1

Add $63$ and $1$ .

$\frac{x \cdot 9 (\sqrt[6]{64} - 2)}{9 + x \cdot - 1}$

Step 3.4.3.2

Rewrite $64$ as $2^{6}$ .

$\frac{x \cdot 9 (\sqrt[6]{2^{6}} - 2)}{9 + x \cdot - 1}$

Step 3.4.3.3

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

$\frac{x \cdot 9 (2 - 2)}{9 + x \cdot - 1}$

Step 3.4.4

Subtract $2$ from $2$ .

$\frac{x \cdot 9 \cdot 0}{9 + x \cdot - 1}$

Step 3.4.5

Combine exponents.

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Step 3.4.5.1

Multiply $0$ by $x$ .

$\frac{0 \cdot 9}{9 + x \cdot - 1}$

Step 3.4.5.2

Multiply $0$ by $9$ .

$\frac{0}{9 + x \cdot - 1}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Move $- 1$ to the left of $x$ .

$\frac{0}{9 - 1 \cdot x}$

Step 3.5.2

Rewrite $- 1 x$ as $- x$ .

$\frac{0}{9 - x}$

Step 3.6

Divide $0$ by $9 - x$ .

$0$