Precalculus Examples

$lim_{x \to 8} \frac{\sqrt{4 x^{2} - 7}}{3 x - 1}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{4 x^{2} - 7}}{lim_{x \to 8} 3 x - 1}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 4 x^{2} - 7}}{lim_{x \to 8} 3 x - 1}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 4 x^{2} - lim_{x \to 8} 7}}{lim_{x \to 8} 3 x - 1}$

Step 4

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

$\frac{\sqrt{4 lim_{x \to 8} x^{2} - lim_{x \to 8} 7}}{lim_{x \to 8} 3 x - 1}$

Step 5

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 7}}{lim_{x \to 8} 3 x - 1}$

Step 6

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7}}{lim_{x \to 8} 3 x - 1}$

Step 7

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7}}{lim_{x \to 8} 3 x - lim_{x \to 8} 1}$

Step 8

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7}}{3 lim_{x \to 8} x - lim_{x \to 8} 1}$

Step 9

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7}}{3 lim_{x \to 8} x - 1 \cdot 1}$

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 $x$ .

$\frac{\sqrt{4 \cdot 8^{2} - 1 \cdot 7}}{3 lim_{x \to 8} x - 1 \cdot 1}$

Step 10.2

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

$\frac{\sqrt{4 \cdot 8^{2} - 1 \cdot 7}}{3 \cdot 8 - 1 \cdot 1}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{\sqrt{4 \cdot 64 - 1 \cdot 7}}{3 \cdot 8 - 1 \cdot 1}$

Step 11.1.2

Multiply $4$ by $64$ .

$\frac{\sqrt{256 - 1 \cdot 7}}{3 \cdot 8 - 1 \cdot 1}$

Step 11.1.3

Multiply $- 1$ by $7$ .

$\frac{\sqrt{256 - 7}}{3 \cdot 8 - 1 \cdot 1}$

Step 11.1.4

Subtract $7$ from $256$ .

$\frac{\sqrt{249}}{3 \cdot 8 - 1 \cdot 1}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $3$ by $8$ .

$\frac{\sqrt{249}}{24 - 1 \cdot 1}$

Step 11.2.2

Multiply $- 1$ by $1$ .

$\frac{\sqrt{249}}{24 - 1}$

Step 11.2.3

Subtract $1$ from $24$ .

$\frac{\sqrt{249}}{23}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{249}}{23}$

Decimal Form:

$0.68607538 \dots$