Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

$3 \frac{8^{\frac{3}{2}}}{4 \cdot 8^{\frac{3}{2}} + 4}$

Step 9

Simplify the answer.

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Step 9.1

Multiply $4 \cdot 8^{\frac{3}{2}}$ .

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Step 9.1.1

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

$3 \frac{8^{\frac{3}{2}}}{2^{2} \cdot 8^{\frac{3}{2}} + 4}$

Step 9.1.2

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

$3 \frac{8^{\frac{3}{2}}}{2^{2} \cdot {(2^{3})}^{\frac{3}{2}} + 4}$

Step 9.1.3

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

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Step 9.1.3.1

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

$3 \frac{8^{\frac{3}{2}}}{2^{2} \cdot 2^{3 (\frac{3}{2})} + 4}$

Step 9.1.3.2

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

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Step 9.1.3.2.1

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

$3 \frac{8^{\frac{3}{2}}}{2^{2} \cdot 2^{\frac{3 \cdot 3}{2}} + 4}$

Step 9.1.3.2.2

Multiply $3$ by $3$ .

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

Step 9.1.4

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

$3 \frac{8^{\frac{3}{2}}}{2^{2 + \frac{9}{2}} + 4}$

Step 9.1.5

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

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

Step 9.1.6

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

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

Step 9.1.7

Combine the numerators over the common denominator.

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

Step 9.1.8

Simplify the numerator.

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Step 9.1.8.1

Multiply $2$ by $2$ .

$3 \frac{8^{\frac{3}{2}}}{2^{\frac{4 + 9}{2}} + 4}$

Step 9.1.8.2

Add $4$ and $9$ .

$3 \frac{8^{\frac{3}{2}}}{2^{\frac{13}{2}} + 4}$

Step 9.2

Combine $3$ and $\frac{8^{\frac{3}{2}}}{2^{\frac{13}{2}} + 4}$ .

$\frac{3 \cdot 8^{\frac{3}{2}}}{2^{\frac{13}{2}} + 4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \cdot 8^{\frac{3}{2}}}{2^{\frac{13}{2}} + 4}$

Decimal Form:

$0.71825721 \dots$