Calculus Examples

$\frac{lim_{x \to 9} 9 x \sqrt[6]{6 x + 10} - 2}{9 - x}$

Step 1

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

$\frac{lim_{x \to 9} 9 x \sqrt[6]{6 x + 10} - lim_{x \to 9} 2}{9 - x}$

Step 2

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 lim_{x \to 9} x \sqrt[6]{6 x + 10} - lim_{x \to 9} 2}{9 - x}$

Step 3

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

$\frac{9 lim_{x \to 9} x \cdot lim_{x \to 9} \sqrt[6]{6 x + 10} - lim_{x \to 9} 2}{9 - x}$

Step 4

Move the limit under the radical sign.

$\frac{9 lim_{x \to 9} x \cdot \sqrt[6]{lim_{x \to 9} 6 x + 10} - lim_{x \to 9} 2}{9 - x}$

Step 5

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

$\frac{9 lim_{x \to 9} x \cdot \sqrt[6]{lim_{x \to 9} 6 x + lim_{x \to 9} 10} - lim_{x \to 9} 2}{9 - x}$

Step 6

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 lim_{x \to 9} x \cdot \sqrt[6]{6 lim_{x \to 9} x + lim_{x \to 9} 10} - lim_{x \to 9} 2}{9 - x}$

Step 7

Evaluate the limit of $10$ which is constant as $x$ approaches $9$ .

$\frac{9 lim_{x \to 9} x \cdot \sqrt[6]{6 lim_{x \to 9} x + 10} - lim_{x \to 9} 2}{9 - x}$

Step 8

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

$\frac{9 lim_{x \to 9} x \cdot \sqrt[6]{6 lim_{x \to 9} x + 10} - 1 \cdot 2}{9 - x}$

Step 9

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 9.1

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

$\frac{9 \cdot 9 \cdot \sqrt[6]{6 lim_{x \to 9} x + 10} - 1 \cdot 2}{9 - x}$

Step 9.2

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

$\frac{9 \cdot 9 \cdot \sqrt[6]{6 \cdot 9 + 10} - 1 \cdot 2}{9 - x}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $9$ by $9$ .

$\frac{81 \cdot \sqrt[6]{6 \cdot 9 + 10} - 1 \cdot 2}{9 - x}$

Step 10.2

Multiply $6$ by $9$ .

$\frac{81 \cdot \sqrt[6]{54 + 10} - 1 \cdot 2}{9 - x}$

Step 10.3

Add $54$ and $10$ .

$\frac{81 \cdot \sqrt[6]{64} - 1 \cdot 2}{9 - x}$

Step 10.4

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

$\frac{81 \cdot \sqrt[6]{2^{6}} - 1 \cdot 2}{9 - x}$

Step 10.5

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

$\frac{81 \cdot 2 - 1 \cdot 2}{9 - x}$

Step 10.6

Multiply $81$ by $2$ .

$\frac{162 - 1 \cdot 2}{9 - x}$

Step 10.7

Multiply $- 1$ by $2$ .

$\frac{162 - 2}{9 - x}$

Step 10.8

Subtract $2$ from $162$ .

$\frac{160}{9 - x}$