Calculus Examples

$lim_{x \to 6} \frac{\sqrt{2 x - 4} - \sqrt{x + 2}}{2 - 12}$

Step 1

Move the term $\frac{1}{2 - 12}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2 - 12} lim_{x \to 6} \sqrt{2 x - 4} - \sqrt{x + 2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{1}{2 - 12} (lim_{x \to 6} \sqrt{2 x - 4} - lim_{x \to 6} \sqrt{x + 2})$

Step 3

Move the limit under the radical sign.

$\frac{1}{2 - 12} (\sqrt{lim_{x \to 6} 2 x - 4} - lim_{x \to 6} \sqrt{x + 2})$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{1}{2 - 12} (\sqrt{lim_{x \to 6} 2 x - lim_{x \to 6} 4} - lim_{x \to 6} \sqrt{x + 2})$

Step 5

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

$\frac{1}{2 - 12} (\sqrt{2 lim_{x \to 6} x - lim_{x \to 6} 4} - lim_{x \to 6} \sqrt{x + 2})$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $6$ .

$\frac{1}{2 - 12} (\sqrt{2 lim_{x \to 6} x - 1 \cdot 4} - lim_{x \to 6} \sqrt{x + 2})$

Step 7

Move the limit under the radical sign.

$\frac{1}{2 - 12} (\sqrt{2 lim_{x \to 6} x - 1 \cdot 4} - \sqrt{lim_{x \to 6} x + 2})$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{1}{2 - 12} (\sqrt{2 lim_{x \to 6} x - 1 \cdot 4} - \sqrt{lim_{x \to 6} x + lim_{x \to 6} 2})$

Step 9

Evaluate the limit of $2$ which is constant as $x$ approaches $6$ .

$\frac{1}{2 - 12} (\sqrt{2 lim_{x \to 6} x - 1 \cdot 4} - \sqrt{lim_{x \to 6} x + 2})$

Step 10

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{1}{2 - 12} (\sqrt{2 \cdot 6 - 1 \cdot 4} - \sqrt{lim_{x \to 6} x + 2})$

Step 10.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{1}{2 - 12} (\sqrt{2 \cdot 6 - 1 \cdot 4} - \sqrt{6 + 2})$

Step 11

Simplify the answer.

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Step 11.1

Subtract $12$ from $2$ .

$\frac{1}{- 10} (\sqrt{2 \cdot 6 - 1 \cdot 4} - \sqrt{6 + 2})$

Step 11.2

Move the negative in front of the fraction.

$- \frac{1}{10} (\sqrt{2 \cdot 6 - 1 \cdot 4} - \sqrt{6 + 2})$

Step 11.3

Simplify each term.

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Step 11.3.1

Multiply $2$ by $6$ .

$- \frac{1}{10} (\sqrt{12 - 1 \cdot 4} - \sqrt{6 + 2})$

Step 11.3.2

Multiply $- 1$ by $4$ .

$- \frac{1}{10} (\sqrt{12 - 4} - \sqrt{6 + 2})$

Step 11.3.3

Subtract $4$ from $12$ .

$- \frac{1}{10} (\sqrt{8} - \sqrt{6 + 2})$

Step 11.3.4

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

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Step 11.3.4.1

Factor $4$ out of $8$ .

$- \frac{1}{10} (\sqrt{4 (2)} - \sqrt{6 + 2})$

Step 11.3.4.2

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

$- \frac{1}{10} (\sqrt{2^{2} \cdot 2} - \sqrt{6 + 2})$

Step 11.3.5

Pull terms out from under the radical.

$- \frac{1}{10} (2 \sqrt{2} - \sqrt{6 + 2})$

Step 11.3.6

Add $6$ and $2$ .

$- \frac{1}{10} (2 \sqrt{2} - \sqrt{8})$

Step 11.3.7

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

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Step 11.3.7.1

Factor $4$ out of $8$ .

$- \frac{1}{10} (2 \sqrt{2} - \sqrt{4 (2)})$

Step 11.3.7.2

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

$- \frac{1}{10} (2 \sqrt{2} - \sqrt{2^{2} \cdot 2})$

Step 11.3.8

Pull terms out from under the radical.

$- \frac{1}{10} (2 \sqrt{2} - (2 \sqrt{2}))$

Step 11.3.9

Multiply $2$ by $- 1$ .

$- \frac{1}{10} (2 \sqrt{2} - 2 \sqrt{2})$

Step 11.4

Subtract $2 \sqrt{2}$ from $2 \sqrt{2}$ .

$- \frac{1}{10} \cdot 0$

Step 11.5

Multiply $- \frac{1}{10} \cdot 0$ .

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Step 11.5.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{10})$

Step 11.5.2

Multiply $0$ by $\frac{1}{10}$ .

$0$