Calculus Examples

$lim_{x \to 0} \frac{1 - \cos^{2022} (x)}{x \sin (x)}$

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.

$\frac{lim_{x \to 0} 1 - \cos^{2022} (x)}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.1.2

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

$\frac{1 - lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.1.3

Move the exponent $2022$ from $\cos^{2022} (x)$ outside the limit using the Limits Power Rule.

$\frac{1 - {(lim_{x \to 0} \cos (x))}^{2022}}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.1.4

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

$\frac{1 - \cos^{2022} (lim_{x \to 0} x)}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.2

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

$\frac{1 - \cos^{2022} (0)}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

$\frac{1 - 1^{2022}}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.3.1.2

One to any power is one.

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} x \sin (x)}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} x \sin (x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x)}$

Step 1.1.3.2

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

$\frac{0}{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x)}$

Step 1.1.3.3

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

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Step 1.1.3.3.1

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

$\frac{0}{0 \cdot \sin (lim_{x \to 0} x)}$

Step 1.1.3.3.2

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

$\frac{0}{0 \cdot \sin (0)}$

Step 1.1.3.4

Simplify the answer.

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Step 1.1.3.4.1

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

$\frac{0}{0 \cdot 0}$

Step 1.1.3.4.2

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 1.1.3.4.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.5

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.

$lim_{x \to 0} \frac{1 - \cos^{2022} (x)}{x \sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos^{2022} (x)]}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos^{2022} (x)]}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $1 - \cos^{2022} (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos^{2022} (x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos^{2022} (x)]}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.3

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

$lim_{x \to 0} \frac{0 + \frac{d}{d x} [- \cos^{2022} (x)]}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \cos^{2022} (x)]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos^{2022} (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos^{2022} (x)]$ .

$lim_{x \to 0} \frac{0 - \frac{d}{d x} [\cos^{2022} (x)]}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2022}$ and $g (x) = \cos (x)$ .

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Step 1.3.4.2.1

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

$lim_{x \to 0} \frac{0 - (\frac{d}{d u} [u^{2022}] \frac{d}{d x} [\cos (x)])}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2022$ .

$lim_{x \to 0} \frac{0 - (2022 u^{2021} \frac{d}{d x} [\cos (x)])}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4.2.3

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

$lim_{x \to 0} \frac{0 - (2022 \cos^{2021} (x) \frac{d}{d x} [\cos (x)])}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4.3

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

$lim_{x \to 0} \frac{0 - (2022 \cos^{2021} (x) (- \sin (x)))}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4.4

Multiply $- 1$ by $2022$ .

$lim_{x \to 0} \frac{0 - (- 2022 \cos^{2021} (x) \sin (x))}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.4.5

Multiply $- 2022$ by $- 1$ .

$lim_{x \to 0} \frac{0 + 2022 \cos^{2021} (x) \sin (x)}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.5

Add $0$ and $2022 \cos^{2021} (x) \sin (x)$ .

$lim_{x \to 0} \frac{2022 \cos^{2021} (x) \sin (x)}{\frac{d}{d x} [x \sin (x)]}$

Step 1.3.6

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 $f (x) = x$ and $g (x) = \sin (x)$ .

$lim_{x \to 0} \frac{2022 \cos^{2021} (x) \sin (x)}{x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x]}$

Step 1.3.7

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

$lim_{x \to 0} \frac{2022 \cos^{2021} (x) \sin (x)}{x \cos (x) + \sin (x) \frac{d}{d x} [x]}$

Step 1.3.8

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

$lim_{x \to 0} \frac{2022 \cos^{2021} (x) \sin (x)}{x \cos (x) + \sin (x) \cdot 1}$

Step 1.3.9

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

$lim_{x \to 0} \frac{2022 \cos^{2021} (x) \sin (x)}{x \cos (x) + \sin (x)}$

Step 2

Move the term $2022$ outside of the limit because it is constant with respect to $x$ .

$2022 lim_{x \to 0} \frac{\cos^{2021} (x) \sin (x)}{x \cos (x) + \sin (x)}$

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.

$2022 \frac{lim_{x \to 0} \cos^{2021} (x) \sin (x)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

$2022 \frac{lim_{x \to 0} \cos^{2021} (x) \cdot lim_{x \to 0} \sin (x)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.2

Move the exponent $2021$ from $\cos^{2021} (x)$ outside the limit using the Limits Power Rule.

$2022 \frac{{(lim_{x \to 0} \cos (x))}^{2021} \cdot lim_{x \to 0} \sin (x)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.3

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

$2022 \frac{\cos^{2021} (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.4

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

$2022 \frac{\cos^{2021} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.5

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

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Step 3.1.2.5.1

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

$2022 \frac{\cos^{2021} (0) \cdot \sin (lim_{x \to 0} x)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.5.2

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

$2022 \frac{\cos^{2021} (0) \cdot \sin (0)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.6

Simplify the answer.

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Step 3.1.2.6.1

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

$2022 \frac{1^{2021} \cdot \sin (0)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.6.2

One to any power is one.

$2022 \frac{1 \cdot \sin (0)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.6.3

Multiply $\sin (0)$ by $1$ .

$2022 \frac{\sin (0)}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.2.6.4

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

$2022 \frac{0}{lim_{x \to 0} x \cos (x) + \sin (x)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

$2022 \frac{0}{lim_{x \to 0} x \cos (x) + lim_{x \to 0} \sin (x)}$

Step 3.1.3.2

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

$2022 \frac{0}{lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} \sin (x)}$

Step 3.1.3.3

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

$2022 \frac{0}{lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + lim_{x \to 0} \sin (x)}$

Step 3.1.3.4

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

$2022 \frac{0}{lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x) + \sin (lim_{x \to 0} x)}$

Step 3.1.3.5

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

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Step 3.1.3.5.1

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

$2022 \frac{0}{0 \cdot \cos (lim_{x \to 0} x) + \sin (lim_{x \to 0} x)}$

Step 3.1.3.5.2

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

$2022 \frac{0}{0 \cdot \cos (0) + \sin (lim_{x \to 0} x)}$

Step 3.1.3.5.3

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

$2022 \frac{0}{0 \cdot \cos (0) + \sin (0)}$

Step 3.1.3.6

Simplify the answer.

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Step 3.1.3.6.1

Simplify each term.

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Step 3.1.3.6.1.1

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

$2022 \frac{0}{0 \cdot 1 + \sin (0)}$

Step 3.1.3.6.1.2

Multiply $0$ by $1$ .

$2022 \frac{0}{0 + \sin (0)}$

Step 3.1.3.6.1.3

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

$2022 \frac{0}{0 + 0}$

Step 3.1.3.6.2

Add $0$ and $0$ .

$2022 (\frac{0}{0})$

Step 3.1.3.6.3

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

Undefined

$2022 (\frac{0}{0})$

Step 3.1.3.7

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

Undefined

$2022 (\frac{0}{0})$

Step 3.1.4

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

Undefined

$2022 (\frac{0}{0})$

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.

$lim_{x \to 0} \frac{\cos^{2021} (x) \sin (x)}{x \cos (x) + \sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\cos^{2021} (x) \sin (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$2022 lim_{x \to 0} \frac{\frac{d}{d x} [\cos^{2021} (x) \sin (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.2

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 $f (x) = \cos^{2021} (x)$ and $g (x) = \sin (x)$ .

$2022 lim_{x \to 0} \frac{\cos^{2021} (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos^{2021} (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.3

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

$2022 lim_{x \to 0} \frac{\cos^{2021} (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos^{2021} (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.4

Multiply $\cos^{2021} (x)$ by $\cos (x)$ by adding the exponents.

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Step 3.3.4.1

Multiply $\cos^{2021} (x)$ by $\cos (x)$ .

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Step 3.3.4.1.1

Raise $\cos (x)$ to the power of $1$ .

$2022 lim_{x \to 0} \frac{\cos^{2021} (x) \cos^{1} (x) + \sin (x) \frac{d}{d x} [\cos^{2021} (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.4.1.2

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

$2022 lim_{x \to 0} \frac{{\cos (x)}^{2021 + 1} + \sin (x) \frac{d}{d x} [\cos^{2021} (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.4.2

Add $2021$ and $1$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) + \sin (x) \frac{d}{d x} [\cos^{2021} (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2021}$ and $g (x) = \cos (x)$ .

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Step 3.3.5.1

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

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) + \sin (x) \frac{d}{d u} [u^{2021}] \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2021$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) + \sin (x) \cdot 2021 u^{2020} \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.5.3

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

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) + \sin (x) \cdot 2021 \cos^{2020} (x) \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.6

Move $2021$ to the left of $\sin (x)$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) + 2021 \cdot \sin (x) \cos^{2020} (x) \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.7

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

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) + 2021 \sin (x) \cos^{2020} (x) \cdot - 1 \sin (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.8

Multiply $- 1$ by $2021$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) - 2021 \sin (x) \cos^{2020} (x) \sin (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.9

Raise $\sin (x)$ to the power of $1$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) - 2021 \sin^{1} (x) \sin (x) \cos^{2020} (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.10

Raise $\sin (x)$ to the power of $1$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) - 2021 \sin^{1} (x) \sin^{1} (x) \cos^{2020} (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.11

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

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) - 2021 {\sin (x)}^{1 + 1} \cos^{2020} (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.12

Add $1$ and $1$ .

$2022 lim_{x \to 0} \frac{\cos^{2022} (x) - 2021 \sin^{2} (x) \cos^{2020} (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.13

Reorder terms.

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{\frac{d}{d x} [x \cos (x) + \sin (x)]}$

Step 3.3.14

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

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{\frac{d}{d x} [x \cos (x)] + \frac{d}{d x} [\sin (x)]}$

Step 3.3.15

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

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Step 3.3.15.1

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 $f (x) = x$ and $g (x) = \cos (x)$ .

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x] + \frac{d}{d x} [\sin (x)]}$

Step 3.3.15.2

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

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{x \cdot - 1 \sin (x) + \cos (x) \frac{d}{d x} [x] + \frac{d}{d x} [\sin (x)]}$

Step 3.3.15.3

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

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{x \cdot - 1 \sin (x) + \cos (x) \cdot 1 + \frac{d}{d x} [\sin (x)]}$

Step 3.3.15.4

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

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{x \cdot - 1 \sin (x) + \cos (x) + \frac{d}{d x} [\sin (x)]}$

Step 3.3.16

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

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{x \cdot - 1 \sin (x) + \cos (x) + \cos (x)}$

Step 3.3.17

Simplify.

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Step 3.3.17.1

Add $\cos (x)$ and $\cos (x)$ .

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{x \cdot - 1 \sin (x) + 2 \cos (x)}$

Step 3.3.17.2

Reorder terms.

$2022 lim_{x \to 0} \frac{- 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{- x \sin (x) + 2 \cos (x)}$

Step 4

Evaluate the limit.

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Step 4.1

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

$2022 \frac{lim_{x \to 0} - 2021 \cos^{2020} (x) \sin^{2} (x) + \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.2

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

$2022 \frac{- lim_{x \to 0} 2021 \cos^{2020} (x) \sin^{2} (x) + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.3

Move the term $2021$ outside of the limit because it is constant with respect to $x$ .

$2022 \frac{- 2021 lim_{x \to 0} \cos^{2020} (x) \sin^{2} (x) + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.4

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

$2022 \frac{- 2021 lim_{x \to 0} \cos^{2020} (x) \cdot lim_{x \to 0} \sin^{2} (x) + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.5

Move the exponent $2020$ from $\cos^{2020} (x)$ outside the limit using the Limits Power Rule.

$2022 \frac{- 2021 {(lim_{x \to 0} \cos (x))}^{2020} \cdot lim_{x \to 0} \sin^{2} (x) + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.6

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin^{2} (x) + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.7

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot {(lim_{x \to 0} \sin (x))}^{2} + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.8

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + lim_{x \to 0} \cos^{2022} (x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.9

Move the exponent $2022$ from $\cos^{2022} (x)$ outside the limit using the Limits Power Rule.

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + {(lim_{x \to 0} \cos (x))}^{2022}}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.10

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{lim_{x \to 0} - x \sin (x) + 2 \cos (x)}$

Step 4.11

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \sin (x) + lim_{x \to 0} 2 \cos (x)}$

Step 4.12

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} 2 \cos (x)}$

Step 4.13

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + lim_{x \to 0} 2 \cos (x)}$

Step 4.14

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 2 lim_{x \to 0} \cos (x)}$

Step 4.15

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

$2022 \frac{- 2021 \cos^{2020} (lim_{x \to 0} x) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}$

Step 5

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

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Step 5.1

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

$2022 \frac{- 2021 \cos^{2020} (0) \cdot \sin^{2} (lim_{x \to 0} x) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}$

Step 5.2

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

$2022 \frac{- 2021 \cos^{2020} (0) \cdot \sin^{2} (0) + \cos^{2022} (lim_{x \to 0} x)}{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}$

Step 5.3

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

$2022 \frac{- 2021 \cos^{2020} (0) \cdot \sin^{2} (0) + \cos^{2022} (0)}{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}$

Step 5.4

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

$2022 \frac{- 2021 \cos^{2020} (0) \cdot \sin^{2} (0) + \cos^{2022} (0)}{0 \cdot \sin (lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}$

Step 5.5

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

$2022 \frac{- 2021 \cos^{2020} (0) \cdot \sin^{2} (0) + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (lim_{x \to 0} x)}$

Step 5.6

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

$2022 \frac{- 2021 \cos^{2020} (0) \cdot \sin^{2} (0) + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$2022 \frac{- 2021 \cdot 1^{2020} \cdot \sin^{2} (0) + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.2

One to any power is one.

$2022 \frac{- 2021 \cdot 1 \cdot \sin^{2} (0) + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.3

Multiply $- 2021$ by $1$ .

$2022 \frac{- 2021 \cdot \sin^{2} (0) + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.4

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

$2022 \frac{- 2021 \cdot 0^{2} + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.5

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

$2022 \frac{- 2021 \cdot 0 + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.6

Multiply $- 2021$ by $0$ .

$2022 \frac{0 + \cos^{2022} (0)}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.7

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

$2022 \frac{0 + 1^{2022}}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.8

One to any power is one.

$2022 \frac{0 + 1}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.1.9

Add $0$ and $1$ .

$2022 \frac{1}{0 \cdot \sin (0) + 2 \cos (0)}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

$2022 \frac{1}{0 \cdot 0 + 2 \cos (0)}$

Step 6.2.2

Multiply $0$ by $0$ .

$2022 \frac{1}{0 + 2 \cos (0)}$

Step 6.2.3

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

$2022 \frac{1}{0 + 2 \cdot 1}$

Step 6.2.4

Multiply $2$ by $1$ .

$2022 \frac{1}{0 + 2}$

Step 6.2.5

Add $0$ and $2$ .

$2022 (\frac{1}{2})$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Factor $2$ out of $2022$ .

$2 (1011) \frac{1}{2}$

Step 6.3.2

Cancel the common factor.

$2 \cdot 1011 \frac{1}{2}$

Step 6.3.3

Rewrite the expression.

$1011$