Calculus Examples

$\frac{lim_{x \to 8} r}{r^{2} - 30 \sqrt[3]{r}}$

Step 1

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

$\frac{r}{r^{2} - 30 \sqrt[3]{r}}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{r}$ as $r^{\frac{1}{3}}$ .

$\frac{r}{r^{2} - 30 r^{\frac{1}{3}}}$

Step 2.1.2

Factor $r^{\frac{1}{3}}$ out of $r^{2} - 30 r^{\frac{1}{3}}$ .

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Step 2.1.2.1

Factor $r^{\frac{1}{3}}$ out of $r^{2}$ .

$\frac{r}{r^{\frac{1}{3}} r^{\frac{5}{3}} - 30 r^{\frac{1}{3}}}$

Step 2.1.2.2

Factor $r^{\frac{1}{3}}$ out of $- 30 r^{\frac{1}{3}}$ .

$\frac{r}{r^{\frac{1}{3}} r^{\frac{5}{3}} + r^{\frac{1}{3}} \cdot - 30}$

Step 2.1.2.3

Factor $r^{\frac{1}{3}}$ out of $r^{\frac{1}{3}} r^{\frac{5}{3}} + r^{\frac{1}{3}} \cdot - 30$ .

$\frac{r}{r^{\frac{1}{3}} (r^{\frac{5}{3}} - 30)}$

Step 2.2

Move $r^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{r \cdot r^{- \frac{1}{3}}}{r^{\frac{5}{3}} - 30}$

Step 2.3

Multiply $r$ by $r^{- \frac{1}{3}}$ by adding the exponents.

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Step 2.3.1

Multiply $r$ by $r^{- \frac{1}{3}}$ .

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Step 2.3.1.1

Raise $r$ to the power of $1$ .

$\frac{r^{1} r^{- \frac{1}{3}}}{r^{\frac{5}{3}} - 30}$

Step 2.3.1.2

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

$\frac{r^{1 - \frac{1}{3}}}{r^{\frac{5}{3}} - 30}$

Step 2.3.2

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

$\frac{r^{\frac{3}{3} - \frac{1}{3}}}{r^{\frac{5}{3}} - 30}$

Step 2.3.3

Combine the numerators over the common denominator.

$\frac{r^{\frac{3 - 1}{3}}}{r^{\frac{5}{3}} - 30}$

Step 2.3.4

Subtract $1$ from $3$ .

$\frac{r^{\frac{2}{3}}}{r^{\frac{5}{3}} - 30}$