Calculus Examples

$lim_{x \to 0} \frac{1 - \cos (x) + \sin (3 x)}{1 - \cos (x) + \tan (2 x)}$

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.

$\frac{lim_{x \to 0} 1 - \cos (x) + \sin (3 x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \sin (3 x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.2

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

$\frac{1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \sin (3 x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.3

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

$\frac{1 - \cos (lim_{x \to 0} x) + lim_{x \to 0} \sin (3 x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.4

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

$\frac{1 - \cos (lim_{x \to 0} x) + \sin (lim_{x \to 0} 3 x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.5

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

$\frac{1 - \cos (lim_{x \to 0} x) + \sin (3 lim_{x \to 0} x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{1 - \cos (0) + \sin (3 lim_{x \to 0} x)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.6.2

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

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

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

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

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

Step 1.1.2.7.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.7.1.3

Multiply $3$ by $0$ .

$\frac{1 - 1 + \sin (0)}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.7.1.4

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

$\frac{1 - 1 + 0}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.7.2

Subtract $1$ from $1$ .

$\frac{0 + 0}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.2.7.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} 1 - \cos (x) + \tan (2 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 0} 1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \tan (2 x)}$

Step 1.1.3.2

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

$\frac{0}{1 - lim_{x \to 0} \cos (x) + lim_{x \to 0} \tan (2 x)}$

Step 1.1.3.3

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

$\frac{0}{1 - \cos (lim_{x \to 0} x) + lim_{x \to 0} \tan (2 x)}$

Step 1.1.3.4

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

$\frac{0}{1 - \cos (lim_{x \to 0} x) + \tan (lim_{x \to 0} 2 x)}$

Step 1.1.3.5

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

$\frac{0}{1 - \cos (lim_{x \to 0} x) + \tan (2 lim_{x \to 0} x)}$

Step 1.1.3.6

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

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Step 1.1.3.6.1

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

$\frac{0}{1 - \cos (0) + \tan (2 lim_{x \to 0} x)}$

Step 1.1.3.6.2

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

$\frac{0}{1 - \cos (0) + \tan (2 \cdot 0)}$

Step 1.1.3.7

Simplify the answer.

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Step 1.1.3.7.1

Simplify each term.

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Step 1.1.3.7.1.1

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

$\frac{0}{1 - 1 \cdot 1 + \tan (2 \cdot 0)}$

Step 1.1.3.7.1.2

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1 + \tan (2 \cdot 0)}$

Step 1.1.3.7.1.3

Multiply $2$ by $0$ .

$\frac{0}{1 - 1 + \tan (0)}$

Step 1.1.3.7.1.4

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

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

Step 1.1.3.7.2

Subtract $1$ from $1$ .

$\frac{0}{0 + 0}$

Step 1.1.3.7.3

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.7.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.8

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.

$lim_{x \to 0} \frac{1 - \cos (x) + \sin (3 x)}{1 - \cos (x) + \tan (2 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (x) + \sin (3 x)]}{\frac{d}{d x} [1 - \cos (x) + \tan (2 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.4

Multiply $\sin (x)$ by $1$ .

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

Step 1.3.5

Evaluate $\frac{d}{d x} [\sin (3 x)]$ .

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Step 1.3.5.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 3 x$ .

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Step 1.3.5.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

$lim_{x \to 0} \frac{0 + \sin (x) + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x]}{\frac{d}{d x} [1 - \cos (x) + \tan (2 x)]}$

Step 1.3.5.1.2

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

$lim_{x \to 0} \frac{0 + \sin (x) + \cos (u) \frac{d}{d x} [3 x]}{\frac{d}{d x} [1 - \cos (x) + \tan (2 x)]}$

Step 1.3.5.1.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 1.3.5.2

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

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

Step 1.3.5.3

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

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

Step 1.3.5.4

Multiply $3$ by $1$ .

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

Step 1.3.5.5

Move $3$ to the left of $\cos (3 x)$ .

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

Step 1.3.6

Add $0$ and $\sin (x)$ .

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

Step 1.3.7

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

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

Step 1.3.8

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

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + \frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\tan (2 x)]}$

Step 1.3.9

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

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Step 1.3.9.1

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

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 - \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [\tan (2 x)]}$

Step 1.3.9.2

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

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 - - \sin (x) + \frac{d}{d x} [\tan (2 x)]}$

Step 1.3.9.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + 1 \sin (x) + \frac{d}{d x} [\tan (2 x)]}$

Step 1.3.9.4

Multiply $\sin (x)$ by $1$ .

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + \sin (x) + \frac{d}{d x} [\tan (2 x)]}$

Step 1.3.10

Evaluate $\frac{d}{d x} [\tan (2 x)]$ .

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Step 1.3.10.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 2 x$ .

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Step 1.3.10.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + \sin (x) + \frac{d}{d u} [\tan (u)] \frac{d}{d x} [2 x]}$

Step 1.3.10.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + \sin (x) + \sec^{2} (u) \frac{d}{d x} [2 x]}$

Step 1.3.10.1.3

Replace all occurrences of $u$ with $2 x$ .

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + \sin (x) + \sec^{2} (2 x) \frac{d}{d x} [2 x]}$

Step 1.3.10.2

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

$lim_{x \to 0} \frac{\sin (x) + 3 \cos (3 x)}{0 + \sin (x) + \sec^{2} (2 x) (2 \frac{d}{d x} [x])}$

Step 1.3.10.3

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

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

Step 1.3.10.4

Multiply $2$ by $1$ .

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

Step 1.3.10.5

Move $2$ to the left of $\sec^{2} (2 x)$ .

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

Step 1.3.11

Simplify.

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Step 1.3.11.1

Add $0$ and $\sin (x)$ .

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

Step 1.3.11.2

Simplify each term.

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Step 1.3.11.2.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

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

Step 1.3.11.2.2

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

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

Step 1.3.11.2.3

One to any power is one.

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

Step 1.3.11.2.4

Combine $2$ and $\frac{1}{\cos^{2} (2 x)}$ .

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

Step 1.4

Combine terms.

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Step 1.4.1

To write $\sin (x)$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (2 x)}{\cos^{2} (2 x)}$ .

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

Step 1.4.2

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

Simplify the answer.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Multiply $2$ by $0$ .

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

Step 4.2.2

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

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

Step 4.2.3

One to any power is one.

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

Multiply $2$ by $0$ .

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

Step 4.3.3

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

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

Step 4.3.4

One to any power is one.

$(\sin (0) + 3 \cos (3 \cdot 0)) \frac{1}{0 \cdot 1 + 2}$

Step 4.3.5

Multiply $0$ by $1$ .

$(\sin (0) + 3 \cos (3 \cdot 0)) \frac{1}{0 + 2}$

Step 4.3.6

Add $0$ and $2$ .

$(\sin (0) + 3 \cos (3 \cdot 0)) \frac{1}{2}$

Step 4.4

Simplify each term.

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Step 4.4.1

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

$(0 + 3 \cos (3 \cdot 0)) \frac{1}{2}$

Step 4.4.2

Multiply $3$ by $0$ .

$(0 + 3 \cos (0)) \frac{1}{2}$

Step 4.4.3

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

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

Step 4.4.4

Multiply $3$ by $1$ .

$(0 + 3) \frac{1}{2}$

Step 4.5

Add $0$ and $3$ .

$3 (\frac{1}{2})$

Step 4.6

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$