Calculus Examples

$lim_{y \to 1} \frac{{(y)}^{\frac{1}{3}} - 1}{y} - 1$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $1$ .

$lim_{y \to 1} \frac{{(y)}^{\frac{1}{3}} - 1}{y} - lim_{y \to 1} 1$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $1$ .

$\frac{lim_{y \to 1} {(y)}^{\frac{1}{3}} - 1}{lim_{y \to 1} y} - lim_{y \to 1} 1$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $1$ .

$\frac{lim_{y \to 1} {(y)}^{\frac{1}{3}} - lim_{y \to 1} 1}{lim_{y \to 1} y} - lim_{y \to 1} 1$

Step 4

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

$\frac{{(lim_{y \to 1} y)}^{\frac{1}{3}} - lim_{y \to 1} 1}{lim_{y \to 1} y} - lim_{y \to 1} 1$

Step 5

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

$\frac{{(lim_{y \to 1} y)}^{\frac{1}{3}} - 1 \cdot 1}{lim_{y \to 1} y} - lim_{y \to 1} 1$

Step 6

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

$\frac{{(lim_{y \to 1} y)}^{\frac{1}{3}} - 1 \cdot 1}{lim_{y \to 1} y} - 1 \cdot 1$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $y$ .

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Step 7.1

Evaluate the limit of $y$ by plugging in $1$ for $y$ .

$\frac{1^{\frac{1}{3}} - 1 \cdot 1}{lim_{y \to 1} y} - 1 \cdot 1$

Step 7.2

Evaluate the limit of $y$ by plugging in $1$ for $y$ .

$\frac{1^{\frac{1}{3}} - 1 \cdot 1}{1} - 1 \cdot 1$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 1$ by $1$ .

$\frac{1^{\frac{1}{3}} - 1}{1} - 1 \cdot 1$

Step 8.2

Simplify each term.

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Step 8.2.1

Divide $1^{\frac{1}{3}} - 1$ by $1$ .

$1^{\frac{1}{3}} - 1 - 1 \cdot 1$

Step 8.2.2

One to any power is one.

$1 - 1 - 1 \cdot 1$

Step 8.2.3

Subtract $1$ from $1$ .

$0 - 1 \cdot 1$

Step 8.2.4

Multiply $- 1$ by $1$ .

$0 - 1$

Step 8.3

Subtract $1$ from $0$ .

$- 1$