Calculus Examples

$lim_{x \to 8} \frac{7 \sqrt{x} + x^{- 9}}{9 x - 8}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 7 \sqrt{x} + x^{- 9}}{lim_{x \to 8} 9 x - 8}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 7 \sqrt{x} + lim_{x \to 8} x^{- 9}}{lim_{x \to 8} 9 x - 8}$

Step 3

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

$\frac{7 lim_{x \to 8} \sqrt{x} + lim_{x \to 8} x^{- 9}}{lim_{x \to 8} 9 x - 8}$

Step 4

Move the limit under the radical sign.

$\frac{7 \sqrt{lim_{x \to 8} x} + lim_{x \to 8} x^{- 9}}{lim_{x \to 8} 9 x - 8}$

Step 5

Move the exponent $- 9$ from $x^{- 9}$ outside the limit using the Limits Power Rule.

$\frac{7 \sqrt{lim_{x \to 8} x} + {(lim_{x \to 8} x)}^{- 9}}{lim_{x \to 8} 9 x - 8}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{7 \sqrt{lim_{x \to 8} x} + {(lim_{x \to 8} x)}^{- 9}}{lim_{x \to 8} 9 x - lim_{x \to 8} 8}$

Step 7

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

$\frac{7 \sqrt{lim_{x \to 8} x} + {(lim_{x \to 8} x)}^{- 9}}{9 lim_{x \to 8} x - lim_{x \to 8} 8}$

Step 8

Evaluate the limit of $8$ which is constant as $x$ approaches $8$ .

$\frac{7 \sqrt{lim_{x \to 8} x} + {(lim_{x \to 8} x)}^{- 9}}{9 lim_{x \to 8} x - 1 \cdot 8}$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{7 \sqrt{8} + {(lim_{x \to 8} x)}^{- 9}}{9 lim_{x \to 8} x - 1 \cdot 8}$

Step 9.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{7 \sqrt{8} + 8^{- 9}}{9 lim_{x \to 8} x - 1 \cdot 8}$

Step 9.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{7 \sqrt{8} + 8^{- 9}}{9 \cdot 8 - 1 \cdot 8}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 10.1.1.1

Factor $4$ out of $8$ .

$\frac{7 \sqrt{4 (2)} + 8^{- 9}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{7 \sqrt{2^{2} \cdot 2} + 8^{- 9}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.2

Pull terms out from under the radical.

$\frac{7 (2 \sqrt{2}) + 8^{- 9}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.3

Multiply $2$ by $7$ .

$\frac{14 \sqrt{2} + 8^{- 9}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.4

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

$\frac{14 \sqrt{2} + \frac{1}{8^{9}}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.5

Raise $8$ to the power of $9$ .

$\frac{14 \sqrt{2} + \frac{1}{134217728}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.6

To write $14 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{134217728}{134217728}$ .

$\frac{14 \sqrt{2} \cdot \frac{134217728}{134217728} + \frac{1}{134217728}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.7

Combine $14 \sqrt{2}$ and $\frac{134217728}{134217728}$ .

$\frac{\frac{14 \sqrt{2} \cdot 134217728}{134217728} + \frac{1}{134217728}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.8

Combine the numerators over the common denominator.

$\frac{\frac{14 \sqrt{2} \cdot 134217728 + 1}{134217728}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.1.9

Multiply $134217728$ by $14$ .

$\frac{\frac{1879048192 \sqrt{2} + 1}{134217728}}{9 \cdot 8 - 1 \cdot 8}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $9$ by $8$ .

$\frac{\frac{1879048192 \sqrt{2} + 1}{134217728}}{72 - 1 \cdot 8}$

Step 10.2.2

Multiply $- 1$ by $8$ .

$\frac{\frac{1879048192 \sqrt{2} + 1}{134217728}}{72 - 8}$

Step 10.2.3

Subtract $8$ from $72$ .

$\frac{\frac{1879048192 \sqrt{2} + 1}{134217728}}{64}$

Step 10.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1879048192 \sqrt{2} + 1}{134217728} \cdot \frac{1}{64}$

Step 10.4

Multiply $\frac{1879048192 \sqrt{2} + 1}{134217728} \cdot \frac{1}{64}$ .

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Step 10.4.1

Multiply $\frac{1879048192 \sqrt{2} + 1}{134217728}$ by $\frac{1}{64}$ .

$\frac{1879048192 \sqrt{2} + 1}{134217728 \cdot 64}$

Step 10.4.2

Multiply $134217728$ by $64$ .

$\frac{1879048192 \sqrt{2} + 1}{8589934592}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{1879048192 \sqrt{2} + 1}{8589934592}$

Decimal Form:

$0.30935921 \dots$