Calculus Examples

$lim_{x \to 8} \frac{(\sqrt{5 x} - \sqrt{5 x - 2}) x}{\sqrt{x}}$

Step 1

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

$\frac{lim_{x \to 8} (\sqrt{5 x} - \sqrt{5 x - 2}) x}{lim_{x \to 8} \sqrt{x}}$

Step 2

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

$\frac{lim_{x \to 8} \sqrt{5 x} - \sqrt{5 x - 2} \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 3

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

$\frac{(lim_{x \to 8} \sqrt{5 x} - lim_{x \to 8} \sqrt{5 x - 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 4

Move the limit under the radical sign.

$\frac{(\sqrt{lim_{x \to 8} 5 x} - lim_{x \to 8} \sqrt{5 x - 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{(\sqrt{5 lim_{x \to 8} x} - lim_{x \to 8} \sqrt{5 x - 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 6

Move the limit under the radical sign.

$\frac{(\sqrt{5 lim_{x \to 8} x} - \sqrt{lim_{x \to 8} 5 x - 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 7

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

$\frac{(\sqrt{5 lim_{x \to 8} x} - \sqrt{lim_{x \to 8} 5 x - lim_{x \to 8} 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 8

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{(\sqrt{5 lim_{x \to 8} x} - \sqrt{5 lim_{x \to 8} x - lim_{x \to 8} 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 9

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{(\sqrt{5 lim_{x \to 8} x} - \sqrt{5 lim_{x \to 8} x - 1 \cdot 2}) \cdot lim_{x \to 8} x}{lim_{x \to 8} \sqrt{x}}$

Step 10

Move the limit under the radical sign.

$\frac{(\sqrt{5 lim_{x \to 8} x} - \sqrt{5 lim_{x \to 8} x - 1 \cdot 2}) \cdot lim_{x \to 8} x}{\sqrt{lim_{x \to 8} x}}$

Step 11

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

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Step 11.1

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

$\frac{(\sqrt{5 \cdot 8} - \sqrt{5 lim_{x \to 8} x - 1 \cdot 2}) \cdot lim_{x \to 8} x}{\sqrt{lim_{x \to 8} x}}$

Step 11.2

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

$\frac{(\sqrt{5 \cdot 8} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot lim_{x \to 8} x}{\sqrt{lim_{x \to 8} x}}$

Step 11.3

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

$\frac{(\sqrt{5 \cdot 8} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot 8}{\sqrt{lim_{x \to 8} x}}$

Step 11.4

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

$\frac{(\sqrt{5 \cdot 8} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot 8}{\sqrt{8}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $5$ by $8$ .

$\frac{(\sqrt{40} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot 8}{\sqrt{8}}$

Step 12.1.2

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 12.1.2.1

Factor $4$ out of $40$ .

$\frac{(\sqrt{4 (10)} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot 8}{\sqrt{8}}$

Step 12.1.2.2

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

$\frac{(\sqrt{2^{2} \cdot 10} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot 8}{\sqrt{8}}$

Step 12.1.3

Pull terms out from under the radical.

$\frac{(2 \sqrt{10} - \sqrt{5 \cdot 8 - 1 \cdot 2}) \cdot 8}{\sqrt{8}}$

Step 12.1.4

Multiply $5$ by $8$ .

$\frac{(2 \sqrt{10} - \sqrt{40 - 1 \cdot 2}) \cdot 8}{\sqrt{8}}$

Step 12.1.5

Multiply $- 1$ by $2$ .

$\frac{(2 \sqrt{10} - \sqrt{40 - 2}) \cdot 8}{\sqrt{8}}$

Step 12.1.6

Subtract $2$ from $40$ .

$\frac{(2 \sqrt{10} - \sqrt{38}) \cdot 8}{\sqrt{8}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

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

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Step 12.2.1.1

Factor $4$ out of $8$ .

$\frac{(2 \sqrt{10} - \sqrt{38}) \cdot 8}{\sqrt{4 (2)}}$

Step 12.2.1.2

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

$\frac{(2 \sqrt{10} - \sqrt{38}) \cdot 8}{\sqrt{2^{2} \cdot 2}}$

Step 12.2.2

Pull terms out from under the radical.

$\frac{(2 \sqrt{10} - \sqrt{38}) \cdot 8}{2 \sqrt{2}}$

Step 12.3

Cancel the common factor of $8$ and $2$ .

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Step 12.3.1

Factor $2$ out of $(2 \sqrt{10} - \sqrt{38}) \cdot 8$ .

$\frac{2 ((2 \sqrt{10} - \sqrt{38}) \cdot 4)}{2 \sqrt{2}}$

Step 12.3.2

Cancel the common factors.

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Step 12.3.2.1

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

$\frac{2 ((2 \sqrt{10} - \sqrt{38}) \cdot 4)}{2 (\sqrt{2})}$

Step 12.3.2.2

Cancel the common factor.

$\frac{2 ((2 \sqrt{10} - \sqrt{38}) \cdot 4)}{2 \sqrt{2}}$

Step 12.3.2.3

Rewrite the expression.

$\frac{(2 \sqrt{10} - \sqrt{38}) \cdot 4}{\sqrt{2}}$

Step 12.4

Move $4$ to the left of $2 \sqrt{10} - \sqrt{38}$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38})}{\sqrt{2}}$

Step 12.5

Multiply $\frac{4 \cdot (2 \sqrt{10} - \sqrt{38})}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38})}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 12.6

Combine and simplify the denominator.

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Step 12.6.1

Multiply $\frac{4 \cdot (2 \sqrt{10} - \sqrt{38})}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 12.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 12.6.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 12.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 12.6.5

Add $1$ and $1$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 12.6.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 12.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 12.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 12.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 12.6.6.4

Cancel the common factor of $2$ .

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Step 12.6.6.4.1

Cancel the common factor.

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 12.6.6.4.2

Rewrite the expression.

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{2^{1}}$

Step 12.6.6.5

Evaluate the exponent.

$\frac{4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{2}$

Step 12.7

Cancel the common factor of $4$ and $2$ .

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Step 12.7.1

Factor $2$ out of $4 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}$ .

$\frac{2 (2 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2})}{2}$

Step 12.7.2

Cancel the common factors.

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Step 12.7.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2})}{2 (1)}$

Step 12.7.2.2

Cancel the common factor.

$\frac{2 (2 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2})}{2 \cdot 1}$

Step 12.7.2.3

Rewrite the expression.

$\frac{2 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}}{1}$

Step 12.7.2.4

Divide $2 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}$ by $1$ .

$2 \cdot (2 \sqrt{10} - \sqrt{38}) \sqrt{2}$

Step 12.8

Apply the distributive property.

$(2 (2 \sqrt{10}) + 2 (- \sqrt{38})) \sqrt{2}$

Step 12.9

Multiply $2$ by $2$ .

$(4 \sqrt{10} + 2 (- \sqrt{38})) \sqrt{2}$

Step 12.10

Multiply $- 1$ by $2$ .

$(4 \sqrt{10} - 2 \sqrt{38}) \sqrt{2}$

Step 12.11

Apply the distributive property.

$4 \sqrt{10} \sqrt{2} - 2 \sqrt{38} \sqrt{2}$

Step 12.12

Multiply $4 \sqrt{10} \sqrt{2}$ .

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Step 12.12.1

Combine using the product rule for radicals.

$4 \sqrt{2 \cdot 10} - 2 \sqrt{38} \sqrt{2}$

Step 12.12.2

Multiply $2$ by $10$ .

$4 \sqrt{20} - 2 \sqrt{38} \sqrt{2}$

Step 12.13

Multiply $- 2 \sqrt{38} \sqrt{2}$ .

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Step 12.13.1

Combine using the product rule for radicals.

$4 \sqrt{20} - 2 \sqrt{2 \cdot 38}$

Step 12.13.2

Multiply $2$ by $38$ .

$4 \sqrt{20} - 2 \sqrt{76}$

Step 12.14

Simplify each term.

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Step 12.14.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 12.14.1.1

Factor $4$ out of $20$ .

$4 \sqrt{4 (5)} - 2 \sqrt{76}$

Step 12.14.1.2

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

$4 \sqrt{2^{2} \cdot 5} - 2 \sqrt{76}$

Step 12.14.2

Pull terms out from under the radical.

$4 (2 \sqrt{5}) - 2 \sqrt{76}$

Step 12.14.3

Multiply $2$ by $4$ .

$8 \sqrt{5} - 2 \sqrt{76}$

Step 12.14.4

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 12.14.4.1

Factor $4$ out of $76$ .

$8 \sqrt{5} - 2 \sqrt{4 (19)}$

Step 12.14.4.2

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

$8 \sqrt{5} - 2 \sqrt{2^{2} \cdot 19}$

Step 12.14.5

Pull terms out from under the radical.

$8 \sqrt{5} - 2 (2 \sqrt{19})$

Step 12.14.6

Multiply $2$ by $- 2$ .

$8 \sqrt{5} - 4 \sqrt{19}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{5} - 4 \sqrt{19}$

Decimal Form:

$0.45294804 \dots$