Calculus Examples

$lim_{x \to - 1} \frac{{(x + 1)}^{2} (x - 1)}{x^{3} + 1}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to - 1} {(x + 1)}^{2} (x - 1)}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{lim_{x \to - 1} {(x + 1)}^{2} \cdot lim_{x \to - 1} x - 1}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.2

Move the exponent $2$ from ${(x + 1)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to - 1} x + 1)}^{2} \cdot lim_{x \to - 1} x - 1}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.3

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

$\frac{{(lim_{x \to - 1} x + lim_{x \to - 1} 1)}^{2} \cdot lim_{x \to - 1} x - 1}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.4

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

$\frac{{(lim_{x \to - 1} x + 1)}^{2} \cdot lim_{x \to - 1} x - 1}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.5

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

$\frac{{(lim_{x \to - 1} x + 1)}^{2} \cdot (lim_{x \to - 1} x - lim_{x \to - 1} 1)}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.6

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

$\frac{{(lim_{x \to - 1} x + 1)}^{2} \cdot (lim_{x \to - 1} x - 1 \cdot 1)}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.7

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

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Step 1.1.2.7.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{{(- 1 + 1)}^{2} \cdot (lim_{x \to - 1} x - 1 \cdot 1)}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.7.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{{(- 1 + 1)}^{2} \cdot (- 1 - 1 \cdot 1)}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Add $- 1$ and $1$ .

$\frac{0^{2} \cdot (- 1 - 1 \cdot 1)}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.8.2

Raising $0$ to any positive power yields $0$ .

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

Step 1.1.2.8.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2.8.4

Subtract $1$ from $- 1$ .

$\frac{0 \cdot - 2}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.2.8.5

Multiply $0$ by $- 2$ .

$\frac{0}{lim_{x \to - 1} x^{3} + 1}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to - 1} x^{3} + lim_{x \to - 1} 1}$

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Raise $- 1$ to the power of $3$ .

$\frac{0}{- 1 + 1}$

Step 1.1.3.3.2

Add $- 1$ and $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to - 1} \frac{{(x + 1)}^{2} (x - 1)}{x^{3} + 1} = lim_{x \to - 1} \frac{\frac{d}{d x} [{(x + 1)}^{2} (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 1} \frac{\frac{d}{d x} [{(x + 1)}^{2} (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x + 1) (x + 1) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.3

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.3.3.1

Apply the distributive property.

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x (x + 1) + 1 (x + 1)) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.3.2

Apply the distributive property.

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

Step 1.3.3.3

Apply the distributive property.

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

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $x$ by $x$ .

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.4.1.2

Multiply $x$ by $1$ .

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x^{2} + x + 1 x + 1 \cdot 1) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.4.1.3

Multiply $x$ by $1$ .

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x^{2} + x + x + 1 \cdot 1) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.4.1.4

Multiply $1$ by $1$ .

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x^{2} + x + x + 1) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.4.2

Add $x$ and $x$ .

$lim_{x \to - 1} \frac{\frac{d}{d x} [(x^{2} + 2 x + 1) (x - 1)]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 2 x + 1$ and $g (x) = x - 1$ .

$lim_{x \to - 1} \frac{(x^{2} + 2 x + 1) \frac{d}{d x} [x - 1] + (x - 1) \frac{d}{d x} [x^{2} + 2 x + 1]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.6

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to - 1} \frac{(x^{2} + 2 x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x^{2} + 2 x + 1]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 1} \frac{(x^{2} + 2 x + 1) (1 + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [x^{2} + 2 x + 1]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to - 1} \frac{(x^{2} + 2 x + 1) (1 + 0) + (x - 1) \frac{d}{d x} [x^{2} + 2 x + 1]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to - 1} \frac{(x^{2} + 2 x + 1) \cdot 1 + (x - 1) \frac{d}{d x} [x^{2} + 2 x + 1]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.10

Multiply $x^{2} + 2 x + 1$ by $1$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) \frac{d}{d x} [x^{2} + 2 x + 1]}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.11

By the Sum Rule, the derivative of $x^{2} + 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1])}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1])}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.13

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1])}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x + 2 \cdot 1 + \frac{d}{d x} [1])}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.15

Multiply $2$ by $1$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x + 2 + \frac{d}{d x} [1])}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.16

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x + 2 + 0)}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.17

Add $2 x + 2$ and $0$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x + 2)}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18

Simplify.

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Step 1.3.18.1

Apply the distributive property.

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + (x - 1) (2 x) + (x - 1) \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.2

Apply the distributive property.

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + x (2 x) - 1 (2 x) + (x - 1) \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.3

Apply the distributive property.

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + x (2 x) - 1 (2 x) + x \cdot 2 - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4

Combine terms.

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Step 1.3.18.4.1

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

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 (x^{1} x) - 1 (2 x) + x \cdot 2 - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.2

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

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 (x^{1} x^{1}) - 1 (2 x) + x \cdot 2 - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.3

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

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{1 + 1} - 1 (2 x) + x \cdot 2 - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.4

Add $1$ and $1$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{2} - 1 (2 x) + x \cdot 2 - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.5

Multiply $2$ by $- 1$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{2} - 2 x + x \cdot 2 - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.6

Move $2$ to the left of $x$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{2} - 2 x + 2 \cdot x - 1 \cdot 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.7

Multiply $- 1$ by $2$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{2} - 2 x + 2 x - 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.8

Add $- 2 x$ and $2 x$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{2} + 0 - 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.9

Add $2 x^{2}$ and $0$ .

$lim_{x \to - 1} \frac{x^{2} + 2 x + 1 + 2 x^{2} - 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.10

Add $x^{2}$ and $2 x^{2}$ .

$lim_{x \to - 1} \frac{3 x^{2} + 2 x + 1 - 2}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.18.4.11

Subtract $2$ from $1$ .

$lim_{x \to - 1} \frac{3 x^{2} + 2 x - 1}{\frac{d}{d x} [x^{3} + 1]}$

Step 1.3.19

By the Sum Rule, the derivative of $x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$ .

$lim_{x \to - 1} \frac{3 x^{2} + 2 x - 1}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]}$

Step 1.3.20

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to - 1} \frac{3 x^{2} + 2 x - 1}{3 x^{2} + \frac{d}{d x} [1]}$

Step 1.3.21

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to - 1} \frac{3 x^{2} + 2 x - 1}{3 x^{2} + 0}$

Step 1.3.22

Add $3 x^{2}$ and $0$ .

$lim_{x \to - 1} \frac{3 x^{2} + 2 x - 1}{3 x^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to - 1} \frac{3 x^{2} + 2 x - 1}{x^{2}}$

Step 2.2

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

$\frac{1}{3} \cdot \frac{lim_{x \to - 1} 3 x^{2} + 2 x - 1}{lim_{x \to - 1} x^{2}}$

Step 2.3

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

$\frac{1}{3} \cdot \frac{lim_{x \to - 1} 3 x^{2} + lim_{x \to - 1} 2 x - lim_{x \to - 1} 1}{lim_{x \to - 1} x^{2}}$

Step 2.4

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

$\frac{1}{3} \cdot \frac{3 lim_{x \to - 1} x^{2} + lim_{x \to - 1} 2 x - lim_{x \to - 1} 1}{lim_{x \to - 1} x^{2}}$

Step 2.5

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

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to - 1} x)}^{2} + lim_{x \to - 1} 2 x - lim_{x \to - 1} 1}{lim_{x \to - 1} x^{2}}$

Step 2.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to - 1} x)}^{2} + 2 lim_{x \to - 1} x - lim_{x \to - 1} 1}{lim_{x \to - 1} x^{2}}$

Step 2.7

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

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to - 1} x)}^{2} + 2 lim_{x \to - 1} x - 1 \cdot 1}{lim_{x \to - 1} x^{2}}$

Step 2.8

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

$\frac{1}{3} \cdot \frac{3 {(lim_{x \to - 1} x)}^{2} + 2 lim_{x \to - 1} x - 1 \cdot 1}{{(lim_{x \to - 1} x)}^{2}}$

Step 3

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

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Step 3.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{1}{3} \cdot \frac{3 {(- 1)}^{2} + 2 lim_{x \to - 1} x - 1 \cdot 1}{{(lim_{x \to - 1} x)}^{2}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{1}{3} \cdot \frac{3 {(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 1}{{(lim_{x \to - 1} x)}^{2}}$

Step 3.3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{1}{3} \cdot \frac{3 {(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 1}{{(- 1)}^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 1$ to the power of $2$ .

$\frac{1}{3} \cdot \frac{3 \cdot 1 + 2 \cdot - 1 - 1 \cdot 1}{{(- 1)}^{2}}$

Step 4.1.2

Multiply $3$ by $1$ .

$\frac{1}{3} \cdot \frac{3 + 2 \cdot - 1 - 1 \cdot 1}{{(- 1)}^{2}}$

Step 4.1.3

Multiply $2$ by $- 1$ .

$\frac{1}{3} \cdot \frac{3 - 2 - 1 \cdot 1}{{(- 1)}^{2}}$

Step 4.1.4

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{3 - 2 - 1}{{(- 1)}^{2}}$

Step 4.1.5

Subtract $2$ from $3$ .

$\frac{1}{3} \cdot \frac{1 - 1}{{(- 1)}^{2}}$

Step 4.1.6

Subtract $1$ from $1$ .

$\frac{1}{3} \cdot \frac{0}{{(- 1)}^{2}}$

Step 4.2

Raise $- 1$ to the power of $2$ .

$\frac{1}{3} \cdot \frac{0}{1}$

Step 4.3

Divide $0$ by $1$ .

$\frac{1}{3} \cdot 0$

Step 4.4

Multiply $\frac{1}{3}$ by $0$ .

$0$