Calculus Examples

$\frac{lim_{x \to 8} 16 x^{\frac{2}{3}}}{4 - x^{\frac{4}{3}}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\frac{16 lim_{x \to 8} x^{\frac{2}{3}}}{4 - x^{\frac{4}{3}}}$

Step 1.2

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

$\frac{16 {(lim_{x \to 8} x)}^{\frac{2}{3}}}{4 - x^{\frac{4}{3}}}$

Step 2

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

$\frac{16 \cdot 8^{\frac{2}{3}}}{4 - x^{\frac{4}{3}}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{2^{4} \cdot 8^{\frac{2}{3}}}{4 - x^{\frac{4}{3}}}$

Step 3.1.2

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

$\frac{2^{4} \cdot {(2^{3})}^{\frac{2}{3}}}{4 - x^{\frac{4}{3}}}$

Step 3.1.3

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

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Step 3.1.3.1

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

$\frac{2^{4} \cdot 2^{3 (\frac{2}{3})}}{4 - x^{\frac{4}{3}}}$

Step 3.1.3.2

Cancel the common factor of $3$ .

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Step 3.1.3.2.1

Cancel the common factor.

$\frac{2^{4} \cdot 2^{3 (\frac{2}{3})}}{4 - x^{\frac{4}{3}}}$

Step 3.1.3.2.2

Rewrite the expression.

$\frac{2^{4} \cdot 2^{2}}{4 - x^{\frac{4}{3}}}$

Step 3.1.4

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

$\frac{2^{4 + 2}}{4 - x^{\frac{4}{3}}}$

Step 3.1.5

Add $4$ and $2$ .

$\frac{2^{6}}{4 - x^{\frac{4}{3}}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

$\frac{2^{6}}{2^{2} - x^{\frac{4}{3}}}$

Step 3.2.2

Rewrite $x^{\frac{4}{3}}$ as ${(x^{\frac{2}{3}})}^{2}$ .

$\frac{2^{6}}{2^{2} - {(x^{\frac{2}{3}})}^{2}}$

Step 3.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x^{\frac{2}{3}}$ .

$\frac{2^{6}}{(2 + x^{\frac{2}{3}}) (2 - x^{\frac{2}{3}})}$

Step 3.3

Raise $2$ to the power of $6$ .

$\frac{64}{(2 + x^{\frac{2}{3}}) (2 - x^{\frac{2}{3}})}$