Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Move the limit under the radical sign.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

Move the limit under the radical sign.

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

Step 10

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

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

Step 11

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 12

Move the limit under the radical sign.

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 13.5

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

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

Step 13.6

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

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

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

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Multiply $2$ by $8$ .

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

Step 14.1.2

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

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

Step 14.1.3

Multiply $4$ by $64$ .

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

Step 14.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 14.1.4.1

Factor $4$ out of $8$ .

$\frac{16 - \sqrt{256 - 8 + 2 \sqrt{4 (2)}}}{2 \sqrt{8} - 8}$

Step 14.1.4.2

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

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

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

Step 14.1.5

Pull terms out from under the radical.

$\frac{16 - \sqrt{256 - 8 + 2 (2 \sqrt{2})}}{2 \sqrt{8} - 8}$

Step 14.1.6

Multiply $2$ by $2$ .

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

Step 14.1.7

Subtract $8$ from $256$ .

$\frac{16 - \sqrt{248 + 4 \sqrt{2}}}{2 \sqrt{8} - 8}$

Step 14.1.8

Rewrite $248 + 4 \sqrt{2}$ as $2^{2} (62 + \sqrt{2})$ .

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Step 14.1.8.1

Factor $4$ out of $248$ .

$\frac{16 - \sqrt{4 (62) + 4 \sqrt{2}}}{2 \sqrt{8} - 8}$

Step 14.1.8.2

Factor $4$ out of $4 \sqrt{2}$ .

$\frac{16 - \sqrt{4 (62) + 4 (\sqrt{2})}}{2 \sqrt{8} - 8}$

Step 14.1.8.3

Factor $4$ out of $4 (62) + 4 (\sqrt{2})$ .

$\frac{16 - \sqrt{4 (62 + \sqrt{2})}}{2 \sqrt{8} - 8}$

Step 14.1.8.4

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

$\frac{16 - \sqrt{2^{2} (62 + \sqrt{2})}}{2 \sqrt{8} - 8}$

$\frac{16 - \sqrt{2^{2} (62 + \sqrt{2})}}{2 \sqrt{8} - 8}$

Step 14.1.9

Pull terms out from under the radical.

$\frac{16 - (2 \sqrt{62 + \sqrt{2}})}{2 \sqrt{8} - 8}$

Step 14.1.10

Multiply $2$ by $- 1$ .

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{2 \sqrt{8} - 8}$

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{2 \sqrt{8} - 8}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 14.2.1.1

Factor $4$ out of $8$ .

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{2 \sqrt{4 (2)} - 8}$

Step 14.2.1.2

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

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{2 \sqrt{2^{2} \cdot 2} - 8}$

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{2 \sqrt{2^{2} \cdot 2} - 8}$

Step 14.2.2

Pull terms out from under the radical.

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{2 (2 \sqrt{2}) - 8}$

Step 14.2.3

Multiply $2$ by $2$ .

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{4 \sqrt{2} - 8}$

$\frac{16 - 2 \sqrt{62 + \sqrt{2}}}{4 \sqrt{2} - 8}$

Step 14.3

Cancel the common factor of $16 - 2 \sqrt{62 + \sqrt{2}}$ and $4 \sqrt{2} - 8$ .

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Step 14.3.1

Factor $2$ out of $16$ .

$\frac{2 (8) - 2 \sqrt{62 + \sqrt{2}}}{4 \sqrt{2} - 8}$

Step 14.3.2

Factor $2$ out of $- 2 \sqrt{62 + \sqrt{2}}$ .

$\frac{2 (8) + 2 (- \sqrt{62 + \sqrt{2}})}{4 \sqrt{2} - 8}$

Step 14.3.3

Factor $2$ out of $2 (8) + 2 (- \sqrt{62 + \sqrt{2}})$ .

$\frac{2 (8 - \sqrt{62 + \sqrt{2}})}{4 \sqrt{2} - 8}$

Step 14.3.4

Cancel the common factors.

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Step 14.3.4.1

Factor $2$ out of $4 \sqrt{2}$ .

$\frac{2 (8 - \sqrt{62 + \sqrt{2}})}{2 (2 \sqrt{2}) - 8}$

Step 14.3.4.2

Factor $2$ out of $- 8$ .

$\frac{2 (8 - \sqrt{62 + \sqrt{2}})}{2 (2 \sqrt{2}) + 2 \cdot - 4}$

Step 14.3.4.3

Factor $2$ out of $2 (2 \sqrt{2}) + 2 (- 4)$ .

$\frac{2 (8 - \sqrt{62 + \sqrt{2}})}{2 (2 \sqrt{2} - 4)}$

Step 14.3.4.4

Cancel the common factor.

$\frac{2 (8 - \sqrt{62 + \sqrt{2}})}{2 (2 \sqrt{2} - 4)}$

Step 14.3.4.5

Rewrite the expression.

$\frac{8 - \sqrt{62 + \sqrt{2}}}{2 \sqrt{2} - 4}$

$\frac{8 - \sqrt{62 + \sqrt{2}}}{2 \sqrt{2} - 4}$

$\frac{8 - \sqrt{62 + \sqrt{2}}}{2 \sqrt{2} - 4}$

Step 14.4

Multiply $\frac{8 - \sqrt{62 + \sqrt{2}}}{2 \sqrt{2} - 4}$ by $\frac{2 \sqrt{2} + 4}{2 \sqrt{2} + 4}$ .

$\frac{8 - \sqrt{62 + \sqrt{2}}}{2 \sqrt{2} - 4} \cdot \frac{2 \sqrt{2} + 4}{2 \sqrt{2} + 4}$

Step 14.5

Multiply $\frac{8 - \sqrt{62 + \sqrt{2}}}{2 \sqrt{2} - 4}$ by $\frac{2 \sqrt{2} + 4}{2 \sqrt{2} + 4}$ .

$\frac{(8 - \sqrt{62 + \sqrt{2}}) (2 \sqrt{2} + 4)}{(2 \sqrt{2} - 4) (2 \sqrt{2} + 4)}$

Step 14.6

Expand the denominator using the FOIL method.

$\frac{(8 - \sqrt{62 + \sqrt{2}}) (2 \sqrt{2} + 4)}{4 {\sqrt{2}}^{2} + 8 \sqrt{2} - 8 \sqrt{2} - 16}$

Step 14.7

Simplify.

$\frac{(8 - \sqrt{62 + \sqrt{2}}) (2 \sqrt{2} + 4)}{- 8}$

Step 14.8

Cancel the common factor of $2 \sqrt{2} + 4$ and $- 8$ .

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Step 14.8.1

Factor $2$ out of $(8 - \sqrt{62 + \sqrt{2}}) (2 \sqrt{2} + 4)$ .

$\frac{2 ((8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2))}{- 8}$

Step 14.8.2

Cancel the common factors.

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Step 14.8.2.1

Factor $2$ out of $- 8$ .

$\frac{2 ((8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2))}{2 (- 4)}$

Step 14.8.2.2

Cancel the common factor.

$\frac{2 ((8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2))}{2 \cdot - 4}$

Step 14.8.2.3

Rewrite the expression.

$\frac{(8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2)}{- 4}$

$\frac{(8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2)}{- 4}$

$\frac{(8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2)}{- 4}$

Step 14.9

Expand $(8 - \sqrt{62 + \sqrt{2}}) (\sqrt{2} + 2)$ using the FOIL Method.

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Step 14.9.1

Apply the distributive property.

$\frac{8 (\sqrt{2} + 2) - \sqrt{62 + \sqrt{2}} (\sqrt{2} + 2)}{- 4}$

Step 14.9.2

Apply the distributive property.

$\frac{8 \sqrt{2} + 8 \cdot 2 - \sqrt{62 + \sqrt{2}} (\sqrt{2} + 2)}{- 4}$

Step 14.9.3

Apply the distributive property.

$\frac{8 \sqrt{2} + 8 \cdot 2 - \sqrt{62 + \sqrt{2}} \sqrt{2} - \sqrt{62 + \sqrt{2}} \cdot 2}{- 4}$

$\frac{8 \sqrt{2} + 8 \cdot 2 - \sqrt{62 + \sqrt{2}} \sqrt{2} - \sqrt{62 + \sqrt{2}} \cdot 2}{- 4}$

Step 14.10

Simplify each term.

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Step 14.10.1

Multiply $8$ by $2$ .

$\frac{8 \sqrt{2} + 16 - \sqrt{62 + \sqrt{2}} \sqrt{2} - \sqrt{62 + \sqrt{2}} \cdot 2}{- 4}$

Step 14.10.2

Combine using the product rule for radicals.

$\frac{8 \sqrt{2} + 16 - \sqrt{2 (62 + \sqrt{2})} - \sqrt{62 + \sqrt{2}} \cdot 2}{- 4}$

Step 14.10.3

Multiply $2$ by $- 1$ .

$\frac{8 \sqrt{2} + 16 - \sqrt{2 (62 + \sqrt{2})} - 2 \sqrt{62 + \sqrt{2}}}{- 4}$

$\frac{8 \sqrt{2} + 16 - \sqrt{2 (62 + \sqrt{2})} - 2 \sqrt{62 + \sqrt{2}}}{- 4}$

Step 14.11

Move the negative in front of the fraction.

$- \frac{8 \sqrt{2} + 16 - \sqrt{2 (62 + \sqrt{2})} - 2 \sqrt{62 + \sqrt{2}}}{4}$

$- \frac{8 \sqrt{2} + 16 - \sqrt{2 (62 + \sqrt{2})} - 2 \sqrt{62 + \sqrt{2}}}{4}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{8 \sqrt{2} + 16 - \sqrt{2 (62 + \sqrt{2})} - 2 \sqrt{62 + \sqrt{2}}}{4}$

Decimal Form:

$- 0.03132183 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$