Calculus Examples

$lim_{x \to 8} \sqrt{\frac{9 x^{2} + x - 3}{(x - 7) (x + 1)}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} \frac{9 x^{2} + x - 3}{(x - 7) (x + 1)}}$

Step 2

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

$\sqrt{\frac{lim_{x \to 8} 9 x^{2} + x - 3}{lim_{x \to 8} (x - 7) (x + 1)}}$

Step 3

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

$\sqrt{\frac{lim_{x \to 8} 9 x^{2} + lim_{x \to 8} x - lim_{x \to 8} 3}{lim_{x \to 8} (x - 7) (x + 1)}}$

Step 4

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{\frac{9 lim_{x \to 8} x^{2} + lim_{x \to 8} x - lim_{x \to 8} 3}{lim_{x \to 8} (x - 7) (x + 1)}}$

Step 5

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

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - lim_{x \to 8} 3}{lim_{x \to 8} (x - 7) (x + 1)}}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - 1 \cdot 3}{lim_{x \to 8} (x - 7) (x + 1)}}$

Step 7

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

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - 1 \cdot 3}{lim_{x \to 8} x - 7 \cdot lim_{x \to 8} x + 1}}$

Step 8

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

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - 1 \cdot 3}{(lim_{x \to 8} x - lim_{x \to 8} 7) \cdot lim_{x \to 8} x + 1}}$

Step 9

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

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - 1 \cdot 3}{(lim_{x \to 8} x - 1 \cdot 7) \cdot lim_{x \to 8} x + 1}}$

Step 10

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

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - 1 \cdot 3}{(lim_{x \to 8} x - 1 \cdot 7) \cdot (lim_{x \to 8} x + lim_{x \to 8} 1)}}$

Step 11

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\sqrt{\frac{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x - 1 \cdot 3}{(lim_{x \to 8} x - 1 \cdot 7) \cdot (lim_{x \to 8} x + 1)}}$

Step 12

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

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Step 12.1

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

$\sqrt{\frac{9 \cdot 8^{2} + lim_{x \to 8} x - 1 \cdot 3}{(lim_{x \to 8} x - 1 \cdot 7) \cdot (lim_{x \to 8} x + 1)}}$

Step 12.2

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

$\sqrt{\frac{9 \cdot 8^{2} + 8 - 1 \cdot 3}{(lim_{x \to 8} x - 1 \cdot 7) \cdot (lim_{x \to 8} x + 1)}}$

Step 12.3

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

$\sqrt{\frac{9 \cdot 8^{2} + 8 - 1 \cdot 3}{(8 - 1 \cdot 7) \cdot (lim_{x \to 8} x + 1)}}$

Step 12.4

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

$\sqrt{\frac{9 \cdot 8^{2} + 8 - 1 \cdot 3}{(8 - 1 \cdot 7) \cdot (8 + 1)}}$

Step 13

Simplify the answer.

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Step 13.1

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

$\sqrt{\frac{9 \cdot 64 + 8 - 1 \cdot 3}{(8 - 1 \cdot 7) \cdot (8 + 1)}}$

Step 13.2

Multiply $9$ by $64$ .

$\sqrt{\frac{576 + 8 - 1 \cdot 3}{(8 - 1 \cdot 7) \cdot (8 + 1)}}$

Step 13.3

Multiply $- 1$ by $3$ .

$\sqrt{\frac{576 + 8 - 3}{(8 - 1 \cdot 7) \cdot (8 + 1)}}$

Step 13.4

Add $576$ and $8$ .

$\sqrt{\frac{584 - 3}{(8 - 1 \cdot 7) \cdot (8 + 1)}}$

Step 13.5

Subtract $3$ from $584$ .

$\sqrt{\frac{581}{(8 - 1 \cdot 7) \cdot (8 + 1)}}$

Step 13.6

Multiply $- 1$ by $7$ .

$\sqrt{\frac{581}{(8 - 7) \cdot (8 + 1)}}$

Step 13.7

Subtract $7$ from $8$ .

$\sqrt{\frac{581}{1 \cdot (8 + 1)}}$

Step 13.8

Multiply $8 + 1$ by $1$ .

$\sqrt{\frac{581}{8 + 1}}$

Step 13.9

Add $8$ and $1$ .

$\sqrt{\frac{581}{9}}$

Step 13.10

Rewrite $\sqrt{\frac{581}{9}}$ as $\frac{\sqrt{581}}{\sqrt{9}}$ .

$\frac{\sqrt{581}}{\sqrt{9}}$

Step 13.11

Simplify the denominator.

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Step 13.11.1

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

$\frac{\sqrt{581}}{\sqrt{3^{2}}}$

Step 13.11.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{581}}{3}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{581}}{3}$

Decimal Form:

$8.03464719 \dots$