Calculus Examples

$lim_{x \to 8} (6 x) - 10 \sqrt{x}$

Step 1

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

$lim_{x \to 8} 6 x - lim_{x \to 8} 10 \sqrt{x}$

Step 2

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

$6 lim_{x \to 8} x - lim_{x \to 8} 10 \sqrt{x}$

Step 3

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to 8} x - 10 lim_{x \to 8} \sqrt{x}$

Step 4

Move the limit under the radical sign.

$6 lim_{x \to 8} x - 10 \sqrt{lim_{x \to 8} x}$

Step 5

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

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Step 5.1

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

$6 \cdot 8 - 10 \sqrt{lim_{x \to 8} x}$

Step 5.2

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

$6 \cdot 8 - 10 \sqrt{8}$

Step 6

Simplify each term.

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Step 6.1

Multiply $6$ by $8$ .

$48 - 10 \sqrt{8}$

Step 6.2

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

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Step 6.2.1

Factor $4$ out of $8$ .

$48 - 10 \sqrt{4 (2)}$

Step 6.2.2

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

$48 - 10 \sqrt{2^{2} \cdot 2}$

Step 6.3

Pull terms out from under the radical.

$48 - 10 (2 \sqrt{2})$

Step 6.4

Multiply $2$ by $- 10$ .

$48 - 20 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$48 - 20 \sqrt{2}$

Decimal Form:

$19.71572875 \dots$