Calculus Examples

$lim_{x \to 8} (\sqrt{9 x^{4} + 8}) - 3 x^{2} - \frac{3}{x^{4}} + 3$

Step 1

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

$lim_{x \to 8} \sqrt{9 x^{4} + 8} - lim_{x \to 8} 3 x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} 9 x^{4} + 8} - lim_{x \to 8} 3 x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 3

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

$\sqrt{lim_{x \to 8} 9 x^{4} + lim_{x \to 8} 8} - lim_{x \to 8} 3 x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 4

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

$\sqrt{9 lim_{x \to 8} x^{4} + lim_{x \to 8} 8} - lim_{x \to 8} 3 x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 5

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 8} - lim_{x \to 8} 3 x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 6

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - lim_{x \to 8} 3 x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 7

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 lim_{x \to 8} x^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 8

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} \frac{3}{x^{4}} + lim_{x \to 8} 3$

Step 9

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} \frac{1}{x^{4}} + lim_{x \to 8} 3$

Step 10

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - 3 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{4}} + lim_{x \to 8} 3$

Step 11

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - 3 \frac{1}{lim_{x \to 8} x^{4}} + lim_{x \to 8} 3$

Step 12

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - 3 \frac{1}{{(lim_{x \to 8} x)}^{4}} + lim_{x \to 8} 3$

Step 13

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

$\sqrt{9 {(lim_{x \to 8} x)}^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - 3 \frac{1}{{(lim_{x \to 8} x)}^{4}} + 3$

Step 14

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

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Step 14.1

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

$\sqrt{9 \cdot 8^{4} + 8} - 3 {(lim_{x \to 8} x)}^{2} - 3 \frac{1}{{(lim_{x \to 8} x)}^{4}} + 3$

Step 14.2

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

$\sqrt{9 \cdot 8^{4} + 8} - 3 \cdot 8^{2} - 3 \frac{1}{{(lim_{x \to 8} x)}^{4}} + 3$

Step 14.3

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

$\sqrt{9 \cdot 8^{4} + 8} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15

Simplify the answer.

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$\sqrt{9 \cdot 4096 + 8} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15.1.2

Multiply $9$ by $4096$ .

$\sqrt{36864 + 8} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15.1.3

Add $36864$ and $8$ .

$\sqrt{36872} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15.1.4

Rewrite $36872$ as $2^{2} \cdot 9218$ .

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Step 15.1.4.1

Factor $4$ out of $36872$ .

$\sqrt{4 (9218)} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15.1.4.2

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

$\sqrt{2^{2} \cdot 9218} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15.1.5

Pull terms out from under the radical.

$2 \sqrt{9218} - 3 \cdot 8^{2} - 3 \frac{1}{8^{4}} + 3$

Step 15.1.6

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

$2 \sqrt{9218} - 3 \cdot 64 - 3 \frac{1}{8^{4}} + 3$

Step 15.1.7

Multiply $- 3$ by $64$ .

$2 \sqrt{9218} - 192 - 3 \frac{1}{8^{4}} + 3$

Step 15.1.8

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

$2 \sqrt{9218} - 192 - 3 (\frac{1}{4096}) + 3$

Step 15.1.9

Combine $- 3$ and $\frac{1}{4096}$ .

$2 \sqrt{9218} - 192 + \frac{- 3}{4096} + 3$

Step 15.1.10

Move the negative in front of the fraction.

$2 \sqrt{9218} - 192 - \frac{3}{4096} + 3$

Step 15.2

To write $- 192$ as a fraction with a common denominator, multiply by $\frac{4096}{4096}$ .

$2 \sqrt{9218} - 192 \cdot \frac{4096}{4096} - \frac{3}{4096} + 3$

Step 15.3

Combine $- 192$ and $\frac{4096}{4096}$ .

$2 \sqrt{9218} + \frac{- 192 \cdot 4096}{4096} - \frac{3}{4096} + 3$

Step 15.4

Combine the numerators over the common denominator.

$2 \sqrt{9218} + \frac{- 192 \cdot 4096 - 3}{4096} + 3$

Step 15.5

Simplify the numerator.

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Step 15.5.1

Multiply $- 192$ by $4096$ .

$2 \sqrt{9218} + \frac{- 786432 - 3}{4096} + 3$

Step 15.5.2

Subtract $3$ from $- 786432$ .

$2 \sqrt{9218} + \frac{- 786435}{4096} + 3$

Step 15.6

Move the negative in front of the fraction.

$2 \sqrt{9218} - \frac{786435}{4096} + 3$

Step 15.7

To write $3$ as a fraction with a common denominator, multiply by $\frac{4096}{4096}$ .

$2 \sqrt{9218} - \frac{786435}{4096} + 3 \cdot \frac{4096}{4096}$

Step 15.8

Combine $3$ and $\frac{4096}{4096}$ .

$2 \sqrt{9218} - \frac{786435}{4096} + \frac{3 \cdot 4096}{4096}$

Step 15.9

Combine the numerators over the common denominator.

$2 \sqrt{9218} + \frac{- 786435 + 3 \cdot 4096}{4096}$

Step 15.10

Simplify the numerator.

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Step 15.10.1

Multiply $3$ by $4096$ .

$2 \sqrt{9218} + \frac{- 786435 + 12288}{4096}$

Step 15.10.2

Add $- 786435$ and $12288$ .

$2 \sqrt{9218} + \frac{- 774147}{4096}$

Step 15.11

Move the negative in front of the fraction.

$2 \sqrt{9218} - \frac{774147}{4096}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{9218} - \frac{774147}{4096}$

Decimal Form:

$3.02009978 \dots$