Calculus Examples

$\frac{lim_{x \to 0} {(1 - 4 x)}^{\frac{1}{3}} - 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} {(1 - 4 x)}^{\frac{1}{3}} - lim_{x \to 0} 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 1.2

Move the exponent $\frac{1}{3}$ from ${(1 - 4 x)}^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} 1 - 4 x)}^{\frac{1}{3}} - lim_{x \to 0} 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} 1 - lim_{x \to 0} 4 x)}^{\frac{1}{3}} - lim_{x \to 0} 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{{(1 - lim_{x \to 0} 4 x)}^{\frac{1}{3}} - lim_{x \to 0} 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 1.5

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

$\frac{{(1 - 4 lim_{x \to 0} x)}^{\frac{1}{3}} - lim_{x \to 0} 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 1.6

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{{(1 - 4 lim_{x \to 0} x)}^{\frac{1}{3}} - 1 \cdot 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{{(1 - 4 \cdot 0)}^{\frac{1}{3}} - 1 \cdot 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 4$ by $0$ .

$\frac{{(1 + 0)}^{\frac{1}{3}} - 1 \cdot 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 3.1.2

Add $1$ and $0$ .

$\frac{1^{\frac{1}{3}} - 1 \cdot 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 3.1.3

One to any power is one.

$\frac{1 - 1 \cdot 1}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 3.1.4

Multiply $- 1$ by $1$ .

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

Step 3.1.5

Subtract $1$ from $1$ .

$\frac{0}{{(1 - 3 x)}^{\frac{1}{5}} - 1}$

Step 3.2

Divide $0$ by ${(1 - 3 x)}^{\frac{1}{5}} - 1$ .

$0$