Calculus Examples

$lim_{x \to 8} \sqrt{x} (1 - \frac{12}{x})$

Step 1

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

$lim_{x \to 8} \sqrt{x} \cdot lim_{x \to 8} 1 - \frac{12}{x}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} x} \cdot lim_{x \to 8} 1 - \frac{12}{x}$

Step 3

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

$\sqrt{lim_{x \to 8} x} \cdot (lim_{x \to 8} 1 - lim_{x \to 8} \frac{12}{x})$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\sqrt{lim_{x \to 8} x} \cdot (1 - lim_{x \to 8} \frac{12}{x})$

Step 5

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{lim_{x \to 8} x} \cdot (1 - 12 lim_{x \to 8} \frac{1}{x})$

Step 6

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

$\sqrt{lim_{x \to 8} x} \cdot (1 - 12 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\sqrt{lim_{x \to 8} x} \cdot (1 - 12 \frac{1}{lim_{x \to 8} x})$

Step 8

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

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Step 8.1

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

$\sqrt{8} \cdot (1 - 12 \frac{1}{lim_{x \to 8} x})$

Step 8.2

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

$\sqrt{8} \cdot (1 - 12 (\frac{1}{8}))$

Step 9

Simplify the answer.

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Step 9.1

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

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Step 9.1.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)} \cdot (1 - 12 (\frac{1}{8}))$

Step 9.1.2

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

$\sqrt{2^{2} \cdot 2} \cdot (1 - 12 (\frac{1}{8}))$

Step 9.2

Pull terms out from under the radical.

$2 \sqrt{2} \cdot (1 - 12 (\frac{1}{8}))$

Step 9.3

Simplify each term.

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Step 9.3.1

Cancel the common factor of $4$ .

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Step 9.3.1.1

Factor $4$ out of $- 12$ .

$2 \sqrt{2} \cdot (1 + 4 (- 3) \frac{1}{8})$

Step 9.3.1.2

Factor $4$ out of $8$ .

$2 \sqrt{2} \cdot (1 + 4 \cdot - 3 \frac{1}{4 \cdot 2})$

Step 9.3.1.3

Cancel the common factor.

$2 \sqrt{2} \cdot (1 + 4 \cdot - 3 \frac{1}{4 \cdot 2})$

Step 9.3.1.4

Rewrite the expression.

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

Step 9.3.2

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

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

Step 9.3.3

Move the negative in front of the fraction.

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

Step 9.4

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

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

Step 9.5

Combine the numerators over the common denominator.

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

Step 9.6

Subtract $3$ from $2$ .

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

Step 9.7

Cancel the common factor of $2$ .

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Step 9.7.1

Factor $2$ out of $2 \sqrt{2}$ .

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

Step 9.7.2

Cancel the common factor.

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

Step 9.7.3

Rewrite the expression.

$\sqrt{2} \cdot - 1$

Step 9.8

Move $- 1$ to the left of $\sqrt{2}$ .

$- 1 \cdot \sqrt{2}$

Step 9.9

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$- \sqrt{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2}$

Decimal Form:

$- 1.41421356 \dots$