Calculus Examples

$lim_{x \to 8} 2 x - 7 - \sqrt{4 x^{2} + 8 x - 5}$

Step 1

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

$lim_{x \to 8} 2 x - lim_{x \to 8} 7 - lim_{x \to 8} \sqrt{4 x^{2} + 8 x - 5}$

Step 2

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

$2 lim_{x \to 8} x - lim_{x \to 8} 7 - lim_{x \to 8} \sqrt{4 x^{2} + 8 x - 5}$

Step 3

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$2 lim_{x \to 8} x - 1 \cdot 7 - lim_{x \to 8} \sqrt{4 x^{2} + 8 x - 5}$

Step 4

Move the limit under the radical sign.

$2 lim_{x \to 8} x - 1 \cdot 7 - \sqrt{lim_{x \to 8} 4 x^{2} + 8 x - 5}$

Step 5

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

$2 lim_{x \to 8} x - 1 \cdot 7 - \sqrt{lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 8 x - lim_{x \to 8} 5}$

Step 6

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

$2 lim_{x \to 8} x - 1 \cdot 7 - \sqrt{4 lim_{x \to 8} x^{2} + lim_{x \to 8} 8 x - lim_{x \to 8} 5}$

Step 7

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

$2 lim_{x \to 8} x - 1 \cdot 7 - \sqrt{4 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 8 x - lim_{x \to 8} 5}$

Step 8

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 8} x - 1 \cdot 7 - \sqrt{4 {(lim_{x \to 8} x)}^{2} + 8 lim_{x \to 8} x - lim_{x \to 8} 5}$

Step 9

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$2 lim_{x \to 8} x - 1 \cdot 7 - \sqrt{4 {(lim_{x \to 8} x)}^{2} + 8 lim_{x \to 8} x - 1 \cdot 5}$

Step 10

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

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Step 10.1

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

$2 \cdot 8 - 1 \cdot 7 - \sqrt{4 {(lim_{x \to 8} x)}^{2} + 8 lim_{x \to 8} x - 1 \cdot 5}$

Step 10.2

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

$2 \cdot 8 - 1 \cdot 7 - \sqrt{4 \cdot 8^{2} + 8 lim_{x \to 8} x - 1 \cdot 5}$

Step 10.3

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

$2 \cdot 8 - 1 \cdot 7 - \sqrt{4 \cdot 8^{2} + 8 \cdot 8 - 1 \cdot 5}$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $2$ by $8$ .

$16 - 1 \cdot 7 - \sqrt{4 \cdot 8^{2} + 8 \cdot 8 - 1 \cdot 5}$

Step 11.1.2

Multiply $- 1$ by $7$ .

$16 - 7 - \sqrt{4 \cdot 8^{2} + 8 \cdot 8 - 1 \cdot 5}$

Step 11.1.3

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

$16 - 7 - \sqrt{4 \cdot 64 + 8 \cdot 8 - 1 \cdot 5}$

Step 11.1.4

Multiply $4$ by $64$ .

$16 - 7 - \sqrt{256 + 8 \cdot 8 - 1 \cdot 5}$

Step 11.1.5

Multiply $8$ by $8$ .

$16 - 7 - \sqrt{256 + 64 - 1 \cdot 5}$

Step 11.1.6

Multiply $- 1$ by $5$ .

$16 - 7 - \sqrt{256 + 64 - 5}$

Step 11.1.7

Add $256$ and $64$ .

$16 - 7 - \sqrt{320 - 5}$

Step 11.1.8

Subtract $5$ from $320$ .

$16 - 7 - \sqrt{315}$

Step 11.1.9

Rewrite $315$ as $3^{2} \cdot 35$ .

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Step 11.1.9.1

Factor $9$ out of $315$ .

$16 - 7 - \sqrt{9 (35)}$

Step 11.1.9.2

Rewrite $9$ as $3^{2}$ .

$16 - 7 - \sqrt{3^{2} \cdot 35}$

Step 11.1.10

Pull terms out from under the radical.

$16 - 7 - (3 \sqrt{35})$

Step 11.1.11

Multiply $3$ by $- 1$ .

$16 - 7 - 3 \sqrt{35}$

Step 11.2

Subtract $7$ from $16$ .

$9 - 3 \sqrt{35}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$9 - 3 \sqrt{35}$

Decimal Form:

$- 8.74823934 \dots$