Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit under the radical sign.

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

Step 5

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

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

Step 6

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

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

Step 7

Move the exponent $\frac{3}{2}$ from $x^{\frac{3}{2}}$ outside the limit using the Limits Power Rule.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

$8 - \frac{8 \cdot \sqrt{8}}{2 \cdot 8^{\frac{3}{2}} + 3 lim_{x \to 8} x - 1 \cdot 5}$

Step 10.5

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

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Simplify the numerator.

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Step 11.1.1.1

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

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Step 11.1.1.1.1

Factor $4$ out of $8$ .

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

Step 11.1.1.1.2

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

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

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

Step 11.1.1.2

Pull terms out from under the radical.

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

Step 11.1.1.3

Multiply $8$ by $2$ .

$8 - \frac{16 \sqrt{2}}{2 \cdot 8^{\frac{3}{2}} + 3 \cdot 8 - 1 \cdot 5}$

$8 - \frac{16 \sqrt{2}}{2 \cdot 8^{\frac{3}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2

Simplify the denominator.

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Step 11.1.2.1

Multiply $2 \cdot 8^{\frac{3}{2}}$ .

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Step 11.1.2.1.1

Rewrite $8$ as $2^{3}$ .

$8 - \frac{16 \sqrt{2}}{2 \cdot {(2^{3})}^{\frac{3}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.2

Multiply the exponents in ${(2^{3})}^{\frac{3}{2}}$ .

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Step 11.1.2.1.2.1

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

$8 - \frac{16 \sqrt{2}}{2 \cdot 2^{3 (\frac{3}{2})} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.2.2

Multiply $3 (\frac{3}{2})$ .

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Step 11.1.2.1.2.2.1

Combine $3$ and $\frac{3}{2}$ .

$8 - \frac{16 \sqrt{2}}{2 \cdot 2^{\frac{3 \cdot 3}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.2.2.2

Multiply $3$ by $3$ .

$8 - \frac{16 \sqrt{2}}{2 \cdot 2^{\frac{9}{2}} + 3 \cdot 8 - 1 \cdot 5}$

$8 - \frac{16 \sqrt{2}}{2 \cdot 2^{\frac{9}{2}} + 3 \cdot 8 - 1 \cdot 5}$

$8 - \frac{16 \sqrt{2}}{2 \cdot 2^{\frac{9}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.3

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

$8 - \frac{16 \sqrt{2}}{2^{1 + \frac{9}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.4

Write $1$ as a fraction with a common denominator.

$8 - \frac{16 \sqrt{2}}{2^{\frac{2}{2} + \frac{9}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.5

Combine the numerators over the common denominator.

$8 - \frac{16 \sqrt{2}}{2^{\frac{2 + 9}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.1.6

Add $2$ and $9$ .

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 3 \cdot 8 - 1 \cdot 5}$

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 3 \cdot 8 - 1 \cdot 5}$

Step 11.1.2.2

Multiply $3$ by $8$ .

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 24 - 1 \cdot 5}$

Step 11.1.2.3

Multiply $- 1$ by $5$ .

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 24 - 5}$

Step 11.1.2.4

Subtract $5$ from $24$ .

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

$8 - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{2^{\frac{11}{2}} + 19}{2^{\frac{11}{2}} + 19}$ .

$\frac{8 (2^{\frac{11}{2}} + 19)}{2^{\frac{11}{2}} + 19} - \frac{16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.3

Combine the numerators over the common denominator.

$\frac{8 (2^{\frac{11}{2}} + 19) - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Apply the distributive property.

$\frac{8 \cdot 2^{\frac{11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2

Multiply $8 \cdot 2^{\frac{11}{2}}$ .

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Step 11.4.2.1

Rewrite $8$ as $2^{3}$ .

$\frac{2^{3} \cdot 2^{\frac{11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2.2

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

$\frac{2^{3 + \frac{11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2^{3 \cdot \frac{2}{2} + \frac{11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2.4

Combine $3$ and $\frac{2}{2}$ .

$\frac{2^{\frac{3 \cdot 2}{2} + \frac{11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2.5

Combine the numerators over the common denominator.

$\frac{2^{\frac{3 \cdot 2 + 11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2.6

Simplify the numerator.

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Step 11.4.2.6.1

Multiply $3$ by $2$ .

$\frac{2^{\frac{6 + 11}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.2.6.2

Add $6$ and $11$ .

$\frac{2^{\frac{17}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

$\frac{2^{\frac{17}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

$\frac{2^{\frac{17}{2}} + 8 \cdot 19 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 11.4.3

Multiply $8$ by $19$ .

$\frac{2^{\frac{17}{2}} + 152 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

$\frac{2^{\frac{17}{2}} + 152 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

$\frac{2^{\frac{17}{2}} + 152 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{17}{2}} + 152 - 16 \sqrt{2}}{2^{\frac{11}{2}} + 19}$

Decimal Form:

$7.64784879 \dots$

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