Calculus Examples

$lim_{x \to 8} \frac{4 x^{2} - 16}{\sqrt{x^{4} + 4 x^{2}} - \sqrt{x^{4} + 16}}$

Step 1

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

$\frac{lim_{x \to 8} 4 x^{2} - 16}{lim_{x \to 8} \sqrt{x^{4} + 4 x^{2}} - \sqrt{x^{4} + 16}}$

Step 2

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

$\frac{lim_{x \to 8} 4 x^{2} - lim_{x \to 8} 16}{lim_{x \to 8} \sqrt{x^{4} + 4 x^{2}} - \sqrt{x^{4} + 16}}$

Step 3

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

$\frac{4 lim_{x \to 8} x^{2} - lim_{x \to 8} 16}{lim_{x \to 8} \sqrt{x^{4} + 4 x^{2}} - \sqrt{x^{4} + 16}}$

Step 4

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 16}{lim_{x \to 8} \sqrt{x^{4} + 4 x^{2}} - \sqrt{x^{4} + 16}}$

Step 5

Evaluate the limit of $16$ which is constant as $x$ approaches $8$ .

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{lim_{x \to 8} \sqrt{x^{4} + 4 x^{2}} - \sqrt{x^{4} + 16}}$

Step 6

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{lim_{x \to 8} \sqrt{x^{4} + 4 x^{2}} - lim_{x \to 8} \sqrt{x^{4} + 16}}$

Step 7

Move the limit under the radical sign.

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{lim_{x \to 8} x^{4} + 4 x^{2}} - lim_{x \to 8} \sqrt{x^{4} + 16}}$

Step 8

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{lim_{x \to 8} x^{4} + lim_{x \to 8} 4 x^{2}} - lim_{x \to 8} \sqrt{x^{4} + 16}}$

Step 9

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 4 x^{2}} - lim_{x \to 8} \sqrt{x^{4} + 16}}$

Step 10

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 lim_{x \to 8} x^{2}} - lim_{x \to 8} \sqrt{x^{4} + 16}}$

Step 11

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 {(lim_{x \to 8} x)}^{2}} - lim_{x \to 8} \sqrt{x^{4} + 16}}$

Step 12

Move the limit under the radical sign.

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 {(lim_{x \to 8} x)}^{2}} - \sqrt{lim_{x \to 8} x^{4} + 16}}$

Step 13

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 {(lim_{x \to 8} x)}^{2}} - \sqrt{lim_{x \to 8} x^{4} + lim_{x \to 8} 16}}$

Step 14

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

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 {(lim_{x \to 8} x)}^{2}} - \sqrt{{(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 16}}$

Step 15

Evaluate the limit of $16$ which is constant as $x$ approaches $8$ .

$\frac{4 {(lim_{x \to 8} x)}^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 {(lim_{x \to 8} x)}^{2}} - \sqrt{{(lim_{x \to 8} x)}^{4} + 16}}$

Step 16

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

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Step 16.1

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

$\frac{4 \cdot 8^{2} - 1 \cdot 16}{\sqrt{{(lim_{x \to 8} x)}^{4} + 4 {(lim_{x \to 8} x)}^{2}} - \sqrt{{(lim_{x \to 8} x)}^{4} + 16}}$

Step 16.2

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

$\frac{4 \cdot 8^{2} - 1 \cdot 16}{\sqrt{8^{4} + 4 {(lim_{x \to 8} x)}^{2}} - \sqrt{{(lim_{x \to 8} x)}^{4} + 16}}$

Step 16.3

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

$\frac{4 \cdot 8^{2} - 1 \cdot 16}{\sqrt{8^{4} + 4 \cdot 8^{2}} - \sqrt{{(lim_{x \to 8} x)}^{4} + 16}}$

Step 16.4

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

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

Step 17

Simplify the answer.

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Step 17.1

Simplify the numerator.

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Step 17.1.1

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

$\frac{4 \cdot 64 - 1 \cdot 16}{\sqrt{8^{4} + 4 \cdot 8^{2}} - \sqrt{8^{4} + 16}}$

Step 17.1.2

Multiply $4$ by $64$ .

$\frac{256 - 1 \cdot 16}{\sqrt{8^{4} + 4 \cdot 8^{2}} - \sqrt{8^{4} + 16}}$

Step 17.1.3

Multiply $- 1$ by $16$ .

$\frac{256 - 16}{\sqrt{8^{4} + 4 \cdot 8^{2}} - \sqrt{8^{4} + 16}}$

Step 17.1.4

Subtract $16$ from $256$ .

$\frac{240}{\sqrt{8^{4} + 4 \cdot 8^{2}} - \sqrt{8^{4} + 16}}$

Step 17.2

Simplify the denominator.

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Step 17.2.1

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

$\frac{240}{\sqrt{4096 + 4 \cdot 8^{2}} - \sqrt{8^{4} + 16}}$

Step 17.2.2

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

$\frac{240}{\sqrt{4096 + 4 \cdot 64} - \sqrt{8^{4} + 16}}$

Step 17.2.3

Multiply $4$ by $64$ .

$\frac{240}{\sqrt{4096 + 256} - \sqrt{8^{4} + 16}}$

Step 17.2.4

Add $4096$ and $256$ .

$\frac{240}{\sqrt{4352} - \sqrt{8^{4} + 16}}$

Step 17.2.5

Rewrite $4352$ as $16^{2} \cdot 17$ .

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Step 17.2.5.1

Factor $256$ out of $4352$ .

$\frac{240}{\sqrt{256 (17)} - \sqrt{8^{4} + 16}}$

Step 17.2.5.2

Rewrite $256$ as $16^{2}$ .

$\frac{240}{\sqrt{16^{2} \cdot 17} - \sqrt{8^{4} + 16}}$

Step 17.2.6

Pull terms out from under the radical.

$\frac{240}{16 \sqrt{17} - \sqrt{8^{4} + 16}}$

Step 17.2.7

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

$\frac{240}{16 \sqrt{17} - \sqrt{4096 + 16}}$

Step 17.2.8

Add $4096$ and $16$ .

$\frac{240}{16 \sqrt{17} - \sqrt{4112}}$

Step 17.2.9

Rewrite $4112$ as $4^{2} \cdot 257$ .

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Step 17.2.9.1

Factor $16$ out of $4112$ .

$\frac{240}{16 \sqrt{17} - \sqrt{16 (257)}}$

Step 17.2.9.2

Rewrite $16$ as $4^{2}$ .

$\frac{240}{16 \sqrt{17} - \sqrt{4^{2} \cdot 257}}$

Step 17.2.10

Pull terms out from under the radical.

$\frac{240}{16 \sqrt{17} - (4 \sqrt{257})}$

Step 17.2.11

Multiply $4$ by $- 1$ .

$\frac{240}{16 \sqrt{17} - 4 \sqrt{257}}$

Step 17.3

Cancel the common factor of $240$ and $16 \sqrt{17} - 4 \sqrt{257}$ .

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Step 17.3.1

Factor $4$ out of $240$ .

$\frac{4 (60)}{16 \sqrt{17} - 4 \sqrt{257}}$

Step 17.3.2

Cancel the common factors.

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Step 17.3.2.1

Factor $4$ out of $16 \sqrt{17}$ .

$\frac{4 (60)}{4 (4 \sqrt{17}) - 4 \sqrt{257}}$

Step 17.3.2.2

Factor $4$ out of $- 4 \sqrt{257}$ .

$\frac{4 (60)}{4 (4 \sqrt{17}) + 4 (- \sqrt{257})}$

Step 17.3.2.3

Factor $4$ out of $4 (4 \sqrt{17}) + 4 (- \sqrt{257})$ .

$\frac{4 (60)}{4 (4 \sqrt{17} - \sqrt{257})}$

Step 17.3.2.4

Cancel the common factor.

$\frac{4 \cdot 60}{4 (4 \sqrt{17} - \sqrt{257})}$

Step 17.3.2.5

Rewrite the expression.

$\frac{60}{4 \sqrt{17} - \sqrt{257}}$

Step 17.4

Multiply $\frac{60}{4 \sqrt{17} - \sqrt{257}}$ by $\frac{4 \sqrt{17} + \sqrt{257}}{4 \sqrt{17} + \sqrt{257}}$ .

$\frac{60}{4 \sqrt{17} - \sqrt{257}} \cdot \frac{4 \sqrt{17} + \sqrt{257}}{4 \sqrt{17} + \sqrt{257}}$

Step 17.5

Multiply $\frac{60}{4 \sqrt{17} - \sqrt{257}}$ by $\frac{4 \sqrt{17} + \sqrt{257}}{4 \sqrt{17} + \sqrt{257}}$ .

$\frac{60 (4 \sqrt{17} + \sqrt{257})}{(4 \sqrt{17} - \sqrt{257}) (4 \sqrt{17} + \sqrt{257})}$

Step 17.6

Expand the denominator using the FOIL method.

$\frac{60 (4 \sqrt{17} + \sqrt{257})}{16 {\sqrt{17}}^{2} + 4 \sqrt{4369} - 4 \sqrt{4369} - {\sqrt{257}}^{2}}$

Step 17.7

Simplify.

$\frac{60 (4 \sqrt{17} + \sqrt{257})}{15}$

Step 17.8

Cancel the common factor of $60$ and $15$ .

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Step 17.8.1

Factor $15$ out of $60 (4 \sqrt{17} + \sqrt{257})$ .

$\frac{15 (4 (4 \sqrt{17} + \sqrt{257}))}{15}$

Step 17.8.2

Cancel the common factors.

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Step 17.8.2.1

Factor $15$ out of $15$ .

$\frac{15 (4 (4 \sqrt{17} + \sqrt{257}))}{15 (1)}$

Step 17.8.2.2

Cancel the common factor.

$\frac{15 (4 (4 \sqrt{17} + \sqrt{257}))}{15 \cdot 1}$

Step 17.8.2.3

Rewrite the expression.

$\frac{4 (4 \sqrt{17} + \sqrt{257})}{1}$

Step 17.8.2.4

Divide $4 (4 \sqrt{17} + \sqrt{257})$ by $1$ .

$4 (4 \sqrt{17} + \sqrt{257})$

Step 17.9

Apply the distributive property.

$4 (4 \sqrt{17}) + 4 \sqrt{257}$

Step 17.10

Multiply $4$ by $4$ .

$16 \sqrt{17} + 4 \sqrt{257}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$16 \sqrt{17} + 4 \sqrt{257}$

Decimal Form:

$130.09456817 \dots$