Calculus Examples

$lim_{x \to 8} \sqrt{4 x^{2} + x} - 2 x$

Step 1

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

$lim_{x \to 8} \sqrt{4 x^{2} + x} - lim_{x \to 8} 2 x$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} 4 x^{2} + x} - lim_{x \to 8} 2 x$

Step 3

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

$\sqrt{lim_{x \to 8} 4 x^{2} + lim_{x \to 8} x} - lim_{x \to 8} 2 x$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{4 lim_{x \to 8} x^{2} + lim_{x \to 8} x} - lim_{x \to 8} 2 x$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\sqrt{4 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x} - lim_{x \to 8} 2 x$

Step 6

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

$\sqrt{4 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x} - 2 lim_{x \to 8} x$

Step 7

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

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Step 7.1

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

$\sqrt{4 \cdot 8^{2} + lim_{x \to 8} x} - 2 lim_{x \to 8} x$

Step 7.2

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

$\sqrt{4 \cdot 8^{2} + 8} - 2 lim_{x \to 8} x$

Step 7.3

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

$\sqrt{4 \cdot 8^{2} + 8} - 2 \cdot 8$

Step 8

Simplify each term.

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Step 8.1

Raise $8$ to the power of $2$ .

$\sqrt{4 \cdot 64 + 8} - 2 \cdot 8$

Step 8.2

Multiply $4$ by $64$ .

$\sqrt{256 + 8} - 2 \cdot 8$

Step 8.3

Add $256$ and $8$ .

$\sqrt{264} - 2 \cdot 8$

Step 8.4

Rewrite $264$ as $2^{2} \cdot 66$ .

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Step 8.4.1

Factor $4$ out of $264$ .

$\sqrt{4 (66)} - 2 \cdot 8$

Step 8.4.2

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

$\sqrt{2^{2} \cdot 66} - 2 \cdot 8$

Step 8.5

Pull terms out from under the radical.

$2 \sqrt{66} - 2 \cdot 8$

Step 8.6

Multiply $- 2$ by $8$ .

$2 \sqrt{66} - 16$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{66} - 16$

Decimal Form:

$0.24807680 \dots$