Calculus Examples

$lim_{x \to 0} {(4 - 3 \cos (x))}^{\frac{1}{x^{2}}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(4 - 3 \cos (x))}^{\frac{1}{x^{2}}}$ as $e^{\ln ({(4 - 3 \cos (x))}^{\frac{1}{x^{2}}})}$ .

$lim_{x \to 0} e^{\ln ({(4 - 3 \cos (x))}^{\frac{1}{x^{2}}})}$

Step 1.2

Expand $\ln ({(4 - 3 \cos (x))}^{\frac{1}{x^{2}}})$ by moving $\frac{1}{x^{2}}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{1}{x^{2}} \ln (4 - 3 \cos (x))}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 0} \frac{1}{x^{2}} \ln (4 - 3 \cos (x))}$

Step 2.2

Combine $\frac{1}{x^{2}}$ and $\ln (4 - 3 \cos (x))$ .

$e^{lim_{x \to 0} \frac{\ln (4 - 3 \cos (x))}{x^{2}}}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$e^{\frac{lim_{x \to 0} \ln (4 - 3 \cos (x))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

Move the limit inside the logarithm.

$e^{\frac{\ln (lim_{x \to 0} 4 - 3 \cos (x))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.1.2

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

$e^{\frac{\ln (lim_{x \to 0} 4 - lim_{x \to 0} 3 \cos (x))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.1.3

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$e^{\frac{\ln (4 - lim_{x \to 0} 3 \cos (x))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.1.4

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

$e^{\frac{\ln (4 - 3 lim_{x \to 0} \cos (x))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.1.5

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

$e^{\frac{\ln (4 - 3 \cos (lim_{x \to 0} x))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.2

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

$e^{\frac{\ln (4 - 3 \cos (0))}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

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

$e^{\frac{\ln (4 - 3 \cdot 1)}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.3.1.2

Multiply $- 3$ by $1$ .

$e^{\frac{\ln (4 - 3)}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.3.2

Subtract $3$ from $4$ .

$e^{\frac{\ln (1)}{lim_{x \to 0} x^{2}}}$

Step 3.1.2.3.3

The natural logarithm of $1$ is $0$ .

$e^{\frac{0}{lim_{x \to 0} x^{2}}}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

$e^{\frac{0}{{(lim_{x \to 0} x)}^{2}}}$

Step 3.1.3.2

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

$e^{\frac{0}{0^{2}}}$

Step 3.1.3.3

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

$e^{\frac{0}{0}}$

Step 3.1.3.4

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

Undefined

$e^{\frac{0}{0}}$

Step 3.1.4

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

Undefined

$e^{\frac{0}{0}}$

Step 3.2

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.

$lim_{x \to 0} \frac{\ln (4 - 3 \cos (x))}{x^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (4 - 3 \cos (x))]}{\frac{d}{d x} [x^{2}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to 0} \frac{\frac{d}{d x} [\ln (4 - 3 \cos (x))]}{\frac{d}{d x} [x^{2}]}}$

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 $f (x) = \ln (x)$ and $g (x) = 4 - 3 \cos (x)$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $4 - 3 \cos (x)$ .

$e^{lim_{x \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [4 - 3 \cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$e^{lim_{x \to 0} \frac{\frac{1}{u} \frac{d}{d x} [4 - 3 \cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.2.3

Replace all occurrences of $u$ with $4 - 3 \cos (x)$ .

$e^{lim_{x \to 0} \frac{\frac{1}{4 - 3 \cos (x)} \frac{d}{d x} [4 - 3 \cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.3

By the Sum Rule, the derivative of $4 - 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 3 \cos (x)]$ .

$e^{lim_{x \to 0} \frac{\frac{1}{4 - 3 \cos (x)} (\frac{d}{d x} [4] + \frac{d}{d x} [- 3 \cos (x)])}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$e^{lim_{x \to 0} \frac{\frac{1}{4 - 3 \cos (x)} (0 + \frac{d}{d x} [- 3 \cos (x)])}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.5

Add $0$ and $\frac{d}{d x} [- 3 \cos (x)]$ .

$e^{lim_{x \to 0} \frac{\frac{1}{4 - 3 \cos (x)} \frac{d}{d x} [- 3 \cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.6

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

$e^{lim_{x \to 0} \frac{\frac{1}{4 - 3 \cos (x)} \cdot - 3 \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.7

Combine $- 3$ and $\frac{1}{4 - 3 \cos (x)}$ .

$e^{lim_{x \to 0} \frac{\frac{- 3}{4 - 3 \cos (x)} \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.8

Move the negative in front of the fraction.

$e^{lim_{x \to 0} \frac{- \frac{3}{4 - 3 \cos (x)} \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.9

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

$e^{lim_{x \to 0} \frac{- \frac{3}{4 - 3 \cos (x)} \cdot - 1 \sin (x)}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.10

Multiply $- 1$ by $- 1$ .

$e^{lim_{x \to 0} \frac{1 \frac{3}{4 - 3 \cos (x)} \sin (x)}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.11

Multiply $\frac{3}{4 - 3 \cos (x)}$ by $1$ .

$e^{lim_{x \to 0} \frac{\frac{3}{4 - 3 \cos (x)} \sin (x)}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.12

Combine $\frac{3}{4 - 3 \cos (x)}$ and $\sin (x)$ .

$e^{lim_{x \to 0} \frac{\frac{3 \sin (x)}{4 - 3 \cos (x)}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.13

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

$e^{lim_{x \to 0} \frac{\frac{3 \sin (x)}{4 - 3 \cos (x)}}{2 x}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{lim_{x \to 0} \frac{3 \sin (x)}{4 - 3 \cos (x)} \cdot \frac{1}{2 x}}$

Step 3.5

Multiply $\frac{3 \sin (x)}{4 - 3 \cos (x)}$ by $\frac{1}{2 x}$ .

$e^{lim_{x \to 0} \frac{3 \sin (x)}{(4 - 3 \cos (x)) \cdot 2 x}}$

Step 4

Move the term $\frac{3}{2}$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\sin (x)}{(4 - 3 \cos (x)) x}}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

$e^{\frac{3}{2} \cdot \frac{lim_{x \to 0} \sin (x)}{lim_{x \to 0} (4 - 3 \cos (x)) x}}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

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

$e^{\frac{3}{2} \cdot \frac{\sin (lim_{x \to 0} x)}{lim_{x \to 0} (4 - 3 \cos (x)) x}}$

Step 5.1.2.2

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

$e^{\frac{3}{2} \cdot \frac{\sin (0)}{lim_{x \to 0} (4 - 3 \cos (x)) x}}$

Step 5.1.2.3

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

$e^{\frac{3}{2} \cdot \frac{0}{lim_{x \to 0} (4 - 3 \cos (x)) x}}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

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

$e^{\frac{3}{2} \cdot \frac{0}{lim_{x \to 0} 4 - 3 \cos (x) \cdot lim_{x \to 0} x}}$

Step 5.1.3.2

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

$e^{\frac{3}{2} \cdot \frac{0}{(lim_{x \to 0} 4 - lim_{x \to 0} 3 \cos (x)) \cdot lim_{x \to 0} x}}$

Step 5.1.3.3

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$e^{\frac{3}{2} \cdot \frac{0}{(4 - lim_{x \to 0} 3 \cos (x)) \cdot lim_{x \to 0} x}}$

Step 5.1.3.4

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

$e^{\frac{3}{2} \cdot \frac{0}{(4 - 3 lim_{x \to 0} \cos (x)) \cdot lim_{x \to 0} x}}$

Step 5.1.3.5

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

$e^{\frac{3}{2} \cdot \frac{0}{(4 - 3 \cos (lim_{x \to 0} x)) \cdot lim_{x \to 0} x}}$

Step 5.1.3.6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1.3.6.1

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

$e^{\frac{3}{2} \cdot \frac{0}{(4 - 3 \cos (0)) \cdot lim_{x \to 0} x}}$

Step 5.1.3.6.2

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

$e^{\frac{3}{2} \cdot \frac{0}{(4 - 3 \cos (0)) \cdot 0}}$

Step 5.1.3.7

Simplify the answer.

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Step 5.1.3.7.1

Simplify each term.

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Step 5.1.3.7.1.1

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

$e^{\frac{3}{2} \cdot \frac{0}{(4 - 3 \cdot 1) \cdot 0}}$

Step 5.1.3.7.1.2

Multiply $- 3$ by $1$ .

$e^{\frac{3}{2} \cdot \frac{0}{(4 - 3) \cdot 0}}$

Step 5.1.3.7.2

Subtract $3$ from $4$ .

$e^{\frac{3}{2} \cdot \frac{0}{1 \cdot 0}}$

Step 5.1.3.7.3

Multiply $0$ by $1$ .

$e^{\frac{3}{2} \cdot \frac{0}{0}}$

Step 5.1.3.7.4

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

Undefined

$e^{\frac{3}{2} \cdot \frac{0}{0}}$

Step 5.1.3.8

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

Undefined

$e^{\frac{3}{2} \cdot \frac{0}{0}}$

Step 5.1.4

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

Undefined

$e^{\frac{3}{2} \cdot \frac{0}{0}}$

Step 5.2

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.

$lim_{x \to 0} \frac{\sin (x)}{(4 - 3 \cos (x)) x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [(4 - 3 \cos (x)) x]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$e^{\frac{3}{2} lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [(4 - 3 \cos (x)) x]}}$

Step 5.3.2

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

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{\frac{d}{d x} [(4 - 3 \cos (x)) x]}}$

Step 5.3.3

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 $f (x) = 4 - 3 \cos (x)$ and $g (x) = x$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{(4 - 3 \cos (x)) \frac{d}{d x} [x] + x \frac{d}{d x} [4 - 3 \cos (x)]}}$

Step 5.3.4

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

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{(4 - 3 \cos (x)) \cdot 1 + x \frac{d}{d x} [4 - 3 \cos (x)]}}$

Step 5.3.5

Multiply $4 - 3 \cos (x)$ by $1$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x \frac{d}{d x} [4 - 3 \cos (x)]}}$

Step 5.3.6

By the Sum Rule, the derivative of $4 - 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 3 \cos (x)]$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x (\frac{d}{d x} [4] + \frac{d}{d x} [- 3 \cos (x)])}}$

Step 5.3.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x (0 + \frac{d}{d x} [- 3 \cos (x)])}}$

Step 5.3.8

Add $0$ and $\frac{d}{d x} [- 3 \cos (x)]$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x \frac{d}{d x} [- 3 \cos (x)]}}$

Step 5.3.9

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

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x \cdot - 3 \frac{d}{d x} [\cos (x)]}}$

Step 5.3.10

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

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x \cdot - 3 \cdot - 1 \sin (x)}}$

Step 5.3.11

Multiply $- 1$ by $- 3$ .

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{4 - 3 \cos (x) + x \cdot 3 \sin (x)}}$

Step 5.3.12

Reorder terms.

$e^{\frac{3}{2} lim_{x \to 0} \frac{\cos (x)}{3 x \sin (x) + 4 - 3 \cos (x)}}$

Step 6

Evaluate the limit.

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Step 6.1

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

$e^{\frac{3}{2} \cdot \frac{lim_{x \to 0} \cos (x)}{lim_{x \to 0} 3 x \sin (x) + 4 - 3 \cos (x)}}$

Step 6.2

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} 3 x \sin (x) + 4 - 3 \cos (x)}}$

Step 6.3

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{lim_{x \to 0} 3 x \sin (x) + lim_{x \to 0} 4 - lim_{x \to 0} 3 \cos (x)}}$

Step 6.4

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{3 lim_{x \to 0} x \sin (x) + lim_{x \to 0} 4 - lim_{x \to 0} 3 \cos (x)}}$

Step 6.5

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{3 lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 4 - lim_{x \to 0} 3 \cos (x)}}$

Step 6.6

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{3 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + lim_{x \to 0} 4 - lim_{x \to 0} 3 \cos (x)}}$

Step 6.7

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{3 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 4 - lim_{x \to 0} 3 \cos (x)}}$

Step 6.8

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{3 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 4 - 3 lim_{x \to 0} \cos (x)}}$

Step 6.9

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

$e^{\frac{3}{2} \cdot \frac{\cos (lim_{x \to 0} x)}{3 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 4 - 3 \cos (lim_{x \to 0} x)}}$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 7.1

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

$e^{\frac{3}{2} \cdot \frac{\cos (0)}{3 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 4 - 3 \cos (lim_{x \to 0} x)}}$

Step 7.2

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

$e^{\frac{3}{2} \cdot \frac{\cos (0)}{3 \cdot 0 \cdot \sin (lim_{x \to 0} x) + 4 - 3 \cos (lim_{x \to 0} x)}}$

Step 7.3

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

$e^{\frac{3}{2} \cdot \frac{\cos (0)}{3 \cdot 0 \cdot \sin (0) + 4 - 3 \cos (lim_{x \to 0} x)}}$

Step 7.4

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

$e^{\frac{3}{2} \cdot \frac{\cos (0)}{3 \cdot 0 \cdot \sin (0) + 4 - 3 \cos (0)}}$

Step 8

Simplify the answer.

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Step 8.1

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

$e^{\frac{3}{2} \cdot \frac{1}{3 \cdot 0 \cdot \sin (0) + 4 - 3 \cos (0)}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $3$ by $0$ .

$e^{\frac{3}{2} \cdot \frac{1}{0 \cdot \sin (0) + 4 - 3 \cos (0)}}$

Step 8.2.2

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

$e^{\frac{3}{2} \cdot \frac{1}{0 \cdot 0 + 4 - 3 \cos (0)}}$

Step 8.2.3

Multiply $0$ by $0$ .

$e^{\frac{3}{2} \cdot \frac{1}{0 + 4 - 3 \cos (0)}}$

Step 8.2.4

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

$e^{\frac{3}{2} \cdot \frac{1}{0 + 4 - 3 \cdot 1}}$

Step 8.2.5

Multiply $- 3$ by $1$ .

$e^{\frac{3}{2} \cdot \frac{1}{0 + 4 - 3}}$

Step 8.2.6

Add $0$ and $4$ .

$e^{\frac{3}{2} \cdot \frac{1}{4 - 3}}$

Step 8.2.7

Subtract $3$ from $4$ .

$e^{\frac{3}{2} \cdot \frac{1}{1}}$

Step 8.3

Cancel the common factor of $1$ .

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Step 8.3.1

Cancel the common factor.

$e^{\frac{3}{2} \cdot \frac{1}{1}}$

Step 8.3.2

Rewrite the expression.

$e^{\frac{3}{2} \cdot 1}$

Step 8.4

Multiply $\frac{3}{2}$ by $1$ .

$e^{\frac{3}{2}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{3}{2}}$

Decimal Form:

$4.48168907 \dots$