Calculus Examples

$lim_{x \to 3} \frac{x - 3}{x^{\frac{4}{3}} - 3 x^{\frac{1}{3}}}$

Step 1

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

$lim_{x \to 3} \frac{x - 3}{x^{\frac{4}{3}} - 3 \sqrt[3]{x}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x - 3}{lim_{x \to 3} x^{\frac{4}{3}} - 3 \sqrt[3]{x}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x - lim_{x \to 3} 3}{lim_{x \to 3} x^{\frac{4}{3}} - 3 \sqrt[3]{x}}$

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x^{\frac{4}{3}} - 3 \sqrt[3]{x}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x^{\frac{4}{3}} - lim_{x \to 3} 3 \sqrt[3]{x}}$

Step 6

Move the exponent $\frac{4}{3}$ from $x^{\frac{4}{3}}$ outside the limit using the Limits Power Rule.

$\frac{lim_{x \to 3} x - 1 \cdot 3}{{(lim_{x \to 3} x)}^{\frac{4}{3}} - lim_{x \to 3} 3 \sqrt[3]{x}}$

Step 7

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

$\frac{lim_{x \to 3} x - 1 \cdot 3}{{(lim_{x \to 3} x)}^{\frac{4}{3}} - 3 lim_{x \to 3} \sqrt[3]{x}}$

Step 8

Move the limit under the radical sign.

$\frac{lim_{x \to 3} x - 1 \cdot 3}{{(lim_{x \to 3} x)}^{\frac{4}{3}} - 3 \sqrt[3]{lim_{x \to 3} x}}$

Step 9

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3 - 1 \cdot 3}{{(lim_{x \to 3} x)}^{\frac{4}{3}} - 3 \sqrt[3]{lim_{x \to 3} x}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3 - 1 \cdot 3}{3^{\frac{4}{3}} - 3 \sqrt[3]{lim_{x \to 3} x}}$

Step 9.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3 - 1 \cdot 3}{3^{\frac{4}{3}} - 3 \sqrt[3]{3}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{3^{\frac{4}{3}} - 3 \sqrt[3]{3}}$

Step 10.1.2

Subtract $3$ from $3$ .

$\frac{0}{3^{\frac{4}{3}} - 3 \sqrt[3]{3}}$

Step 10.2

Factor $- 1$ out of $3^{\frac{4}{3}}$ .

$\frac{0}{- 1 (- 3^{\frac{4}{3}}) - 3 \sqrt[3]{3}}$

Step 10.3

Factor $- 1$ out of $- 3 \sqrt[3]{3}$ .

$\frac{0}{- 1 (- 3^{\frac{4}{3}}) - (3 \sqrt[3]{3})}$

Step 10.4

Factor $- 1$ out of $- 1 (- 3^{\frac{4}{3}}) - (3 \sqrt[3]{3})$ .

$\frac{0}{- 1 (- 3^{\frac{4}{3}} + 3 \sqrt[3]{3})}$

Step 10.5

Factor $3$ out of $0$ .

$\frac{3 (0)}{- 1 (- 3^{\frac{4}{3}} + 3 \sqrt[3]{3})}$

Step 10.6

Cancel the common factors.

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Step 10.6.1

Factor $3$ out of $- 1 (- 3^{\frac{4}{3}} + 3 \sqrt[3]{3})$ .

$\frac{3 (0)}{3 (- 1 (- 3^{\frac{1}{3}} + \sqrt[3]{3}))}$

Step 10.6.2

Cancel the common factor.

$\frac{3 \cdot 0}{3 (- 1 (- 3^{\frac{1}{3}} + \sqrt[3]{3}))}$

Step 10.6.3

Rewrite the expression.

$\frac{0}{- 1 (- 3^{\frac{1}{3}} + \sqrt[3]{3})}$

Step 10.7

Divide $0$ by $- 1 (- 3^{\frac{1}{3}} + \sqrt[3]{3})$ .

$0$