Calculus Examples

$lim_{x \to 8} \sqrt{x^{2} - m x} - \sqrt{x^{2} - n x}$

Step 1

Evaluate the limit of $\sqrt{x^{2} - m x} - \sqrt{x^{2} - n x}$ by plugging in $8$ for $x$ .

$\sqrt{8^{2} - m \cdot 8} - \sqrt{8^{2} - n \cdot 8}$

Step 2

Simplify each term.

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Step 2.1

Factor $8$ out of $8^{2} - m \cdot 8$ .

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Step 2.1.1

Factor $8$ out of $8^{2}$ .

$\sqrt{8 \cdot 8 - m \cdot 8} - \sqrt{8^{2} - n \cdot 8}$

Step 2.1.2

Factor $8$ out of $- m \cdot 8$ .

$\sqrt{8 \cdot 8 + 8 \cdot (- m)} - \sqrt{8^{2} - n \cdot 8}$

Step 2.1.3

Factor $8$ out of $8 \cdot 8 + 8 \cdot (- m)$ .

$\sqrt{8 (8 - m)} - \sqrt{8^{2} - n \cdot 8}$

Step 2.2

Rewrite $8 (8 - m)$ as $2^{2} \cdot (2 (8 - m))$ .

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Step 2.2.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) (8 - m)} - \sqrt{8^{2} - n \cdot 8}$

Step 2.2.2

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

$\sqrt{2^{2} \cdot 2 (8 - m)} - \sqrt{8^{2} - n \cdot 8}$

Step 2.2.3

Add parentheses.

$\sqrt{2^{2} \cdot (2 (8 - m))} - \sqrt{8^{2} - n \cdot 8}$

Step 2.3

Pull terms out from under the radical.

$2 \sqrt{2 (8 - m)} - \sqrt{8^{2} - n \cdot 8}$

Step 2.4

Factor $8$ out of $8^{2} - n \cdot 8$ .

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Step 2.4.1

Factor $8$ out of $8^{2}$ .

$2 \sqrt{2 (8 - m)} - \sqrt{8 \cdot 8 - n \cdot 8}$

Step 2.4.2

Factor $8$ out of $- n \cdot 8$ .

$2 \sqrt{2 (8 - m)} - \sqrt{8 \cdot 8 + 8 \cdot (- n)}$

Step 2.4.3

Factor $8$ out of $8 \cdot 8 + 8 \cdot (- n)$ .

$2 \sqrt{2 (8 - m)} - \sqrt{8 (8 - n)}$

Step 2.5

Rewrite $8 (8 - n)$ as $2^{2} \cdot (2 (8 - n))$ .

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Step 2.5.1

Factor $4$ out of $8$ .

$2 \sqrt{2 (8 - m)} - \sqrt{4 (2) (8 - n)}$

Step 2.5.2

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

$2 \sqrt{2 (8 - m)} - \sqrt{2^{2} \cdot 2 (8 - n)}$

Step 2.5.3

Add parentheses.

$2 \sqrt{2 (8 - m)} - \sqrt{2^{2} \cdot (2 (8 - n))}$

Step 2.6

Pull terms out from under the radical.

$2 \sqrt{2 (8 - m)} - (2 \sqrt{2 (8 - n)})$

Step 2.7

Multiply $2$ by $- 1$ .

$2 \sqrt{2 (8 - m)} - 2 \sqrt{2 (8 - n)}$