Calculus Examples

$lim_{x \to 1} \frac{3 e^{x - 1} - 3}{3 \cos (x - 1) - 3 x^{3}}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 1} 3 e^{x - 1} - 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

$\frac{lim_{x \to 1} 3 e^{x - 1} - lim_{x \to 1} 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.1.2

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

$\frac{3 lim_{x \to 1} e^{x - 1} - lim_{x \to 1} 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.1.3

Move the limit into the exponent.

$\frac{3 e^{lim_{x \to 1} x - 1} - lim_{x \to 1} 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.1.4

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

$\frac{3 e^{lim_{x \to 1} x - lim_{x \to 1} 1} - lim_{x \to 1} 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.1.5

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

$\frac{3 e^{lim_{x \to 1} x - 1 \cdot 1} - lim_{x \to 1} 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.1.6

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{3 e^{lim_{x \to 1} x - 1 \cdot 1} - 1 \cdot 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.2

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

$\frac{3 e^{1 - 1 \cdot 1} - 1 \cdot 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $- 1$ by $1$ .

$\frac{3 e^{1 - 1} - 1 \cdot 3}{lim_{x \to 1} 3 \cos (x - 1) - 3 x^{3}}$

Step 1.2.3.1.2

Subtract $1$ from $1$ .

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

Step 1.2.3.1.3

Anything raised to $0$ is $1$ .

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

Step 1.2.3.1.4

Multiply $3$ by $1$ .

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

Step 1.2.3.1.5

Multiply $- 1$ by $3$ .

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

Step 1.2.3.2

Subtract $3$ from $3$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Move the limit inside the trig function because cosine is continuous.

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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

Step 1.3.9

Simplify the answer.

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Step 1.3.9.1

Simplify each term.

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Step 1.3.9.1.1

Multiply $- 1$ by $1$ .

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

Step 1.3.9.1.2

Subtract $1$ from $1$ .

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

Step 1.3.9.1.3

The exact value of $\cos (0)$ is $1$ .

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

Step 1.3.9.1.4

Multiply $3$ by $1$ .

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

Step 1.3.9.1.5

One to any power is one.

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

Step 1.3.9.1.6

Multiply $- 3$ by $1$ .

$\frac{0}{3 - 3}$

Step 1.3.9.2

Subtract $3$ from $3$ .

$\frac{0}{0}$

Step 1.3.9.3

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

Undefined

$\frac{0}{0}$

Step 1.3.10

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Evaluate $\frac{d}{d x} [3 e^{x - 1}]$ .

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Step 3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x - 1}$ with respect to $x$ is $3 \frac{d}{d x} [e^{x - 1}]$ .

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

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x - 1$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 3.3.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Add $1$ and $0$ .

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

Step 3.3.7

Multiply $e^{x - 1}$ by $1$ .

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

Step 3.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.5

Add $3 e^{x - 1}$ and $0$ .

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

Step 3.6

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

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

Step 3.7

Evaluate $\frac{d}{d x} [3 \cos (x - 1)]$ .

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Step 3.7.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (x - 1)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (x - 1)]$ .

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

Step 3.7.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x - 1$ .

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Step 3.7.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 3.7.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

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

Step 3.7.2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.7.5

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

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

Step 3.7.6

Add $1$ and $0$ .

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

Step 3.7.7

Multiply $- 1$ by $1$ .

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

Step 3.7.8

Multiply $- 1$ by $3$ .

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

Step 3.8

Evaluate $\frac{d}{d x} [- 3 x^{3}]$ .

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Step 3.8.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{3}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{3}]$ .

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

Step 3.8.2

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

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

Step 3.8.3

Multiply $3$ by $- 3$ .

$lim_{x \to 1} \frac{3 e^{x - 1}}{- 3 \sin (x - 1) - 9 x^{2}}$

Step 3.9

Reorder terms.

$lim_{x \to 1} \frac{3 e^{x - 1}}{- 9 x^{2} - 3 \sin (x - 1)}$

Step 4

Cancel the common factor of $3$ and $- 9 x^{2} - 3 \sin (x - 1)$ .

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Step 4.1

Factor $3$ out of $3 e^{x - 1}$ .

$lim_{x \to 1} \frac{3 (e^{x - 1})}{- 9 x^{2} - 3 \sin (x - 1)}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $- 9 x^{2}$ .

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

Step 4.2.2

Factor $3$ out of $- 3 \sin (x - 1)$ .

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

Step 4.2.3

Factor $3$ out of $3 (- 3 x^{2}) + 3 (- \sin (x - 1))$ .

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

Step 4.2.4

Cancel the common factor.

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

Step 4.2.5

Rewrite the expression.

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

Step 5

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

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

Step 6

Move the limit into the exponent.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Move the limit inside the trig function because sine is continuous.

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

Step 13

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

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

Step 14

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

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

Step 15

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

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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

Step 16

Simplify the answer.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Multiply $- 1$ by $1$ .

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

Step 16.1.2

Subtract $1$ from $1$ .

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

Step 16.1.3

Anything raised to $0$ is $1$ .

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

Step 16.2

Simplify the denominator.

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Step 16.2.1

One to any power is one.

$\frac{1}{- 3 \cdot 1 - \sin (1 - 1 \cdot 1)}$

Step 16.2.2

Multiply $- 3$ by $1$ .

$\frac{1}{- 3 - \sin (1 - 1 \cdot 1)}$

Step 16.2.3

Multiply $- 1$ by $1$ .

$\frac{1}{- 3 - \sin (1 - 1)}$

Step 16.2.4

Subtract $1$ from $1$ .

$\frac{1}{- 3 - \sin (0)}$

Step 16.2.5

The exact value of $\sin (0)$ is $0$ .

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

Step 16.2.6

Multiply $- 1$ by $0$ .

$\frac{1}{- 3 + 0}$

Step 16.2.7

Add $- 3$ and $0$ .

$\frac{1}{- 3}$

Step 16.3

Move the negative in front of the fraction.

$- \frac{1}{3}$