Calculus Examples

$lim_{x \to 8} \frac{\sqrt{27 x^{4} + 3 x - 5}}{3 x^{2} - 10}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{27 x^{4} + 3 x - 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 27 x^{4} + 3 x - 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 27 x^{4} + lim_{x \to 8} 3 x - lim_{x \to 8} 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 4

Move the term $27$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{27 lim_{x \to 8} x^{4} + lim_{x \to 8} 3 x - lim_{x \to 8} 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 3 x - lim_{x \to 8} 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 6

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

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + 3 lim_{x \to 8} x - lim_{x \to 8} 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 7

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

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + 3 lim_{x \to 8} x - 1 \cdot 5}}{lim_{x \to 8} 3 x^{2} - 10}$

Step 8

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

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + 3 lim_{x \to 8} x - 1 \cdot 5}}{lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 10}$

Step 9

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

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + 3 lim_{x \to 8} x - 1 \cdot 5}}{3 lim_{x \to 8} x^{2} - lim_{x \to 8} 10}$

Step 10

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

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + 3 lim_{x \to 8} x - 1 \cdot 5}}{3 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 10}$

Step 11

Evaluate the limit of $10$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{27 {(lim_{x \to 8} x)}^{4} + 3 lim_{x \to 8} x - 1 \cdot 5}}{3 {(lim_{x \to 8} x)}^{2} - 1 \cdot 10}$

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 $x$ .

$\frac{\sqrt{27 \cdot 8^{4} + 3 lim_{x \to 8} x - 1 \cdot 5}}{3 {(lim_{x \to 8} x)}^{2} - 1 \cdot 10}$

Step 12.2

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

$\frac{\sqrt{27 \cdot 8^{4} + 3 \cdot 8 - 1 \cdot 5}}{3 {(lim_{x \to 8} x)}^{2} - 1 \cdot 10}$

Step 12.3

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

$\frac{\sqrt{27 \cdot 8^{4} + 3 \cdot 8 - 1 \cdot 5}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Raise $8$ to the power of $4$ .

$\frac{\sqrt{27 \cdot 4096 + 3 \cdot 8 - 1 \cdot 5}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13.1.2

Multiply $27$ by $4096$ .

$\frac{\sqrt{110592 + 3 \cdot 8 - 1 \cdot 5}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13.1.3

Multiply $3$ by $8$ .

$\frac{\sqrt{110592 + 24 - 1 \cdot 5}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13.1.4

Multiply $- 1$ by $5$ .

$\frac{\sqrt{110592 + 24 - 5}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13.1.5

Add $110592$ and $24$ .

$\frac{\sqrt{110616 - 5}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13.1.6

Subtract $5$ from $110616$ .

$\frac{\sqrt{110611}}{3 \cdot 8^{2} - 1 \cdot 10}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{\sqrt{110611}}{3 \cdot 64 - 1 \cdot 10}$

Step 13.2.2

Multiply $3$ by $64$ .

$\frac{\sqrt{110611}}{192 - 1 \cdot 10}$

Step 13.2.3

Multiply $- 1$ by $10$ .

$\frac{\sqrt{110611}}{192 - 10}$

Step 13.2.4

Subtract $10$ from $192$ .

$\frac{\sqrt{110611}}{182}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{110611}}{182}$

Decimal Form:

$1.82737538 \dots$