Calculus Examples

$lim_{x \to 0} \frac{1 + (x - 1) \cos (x)}{4 x}$

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to 0} \frac{1 + (x - 1) \cos (x)}{x}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{4} \cdot \frac{lim_{x \to 0} 1 + (x - 1) \cos (x)}{lim_{x \to 0} x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

$\frac{1}{4} \cdot \frac{lim_{x \to 0} 1 + lim_{x \to 0} (x - 1) \cos (x)}{lim_{x \to 0} x}$

Step 2.1.2.2

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

$\frac{1}{4} \cdot \frac{1 + lim_{x \to 0} (x - 1) \cos (x)}{lim_{x \to 0} x}$

Step 2.1.2.3

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

$\frac{1}{4} \cdot \frac{1 + lim_{x \to 0} x - 1 \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 2.1.2.4

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

$\frac{1}{4} \cdot \frac{1 + (lim_{x \to 0} x - lim_{x \to 0} 1) \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 2.1.2.5

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

$\frac{1}{4} \cdot \frac{1 + (lim_{x \to 0} x - 1 \cdot 1) \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 2.1.2.6

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

$\frac{1}{4} \cdot \frac{1 + (lim_{x \to 0} x - 1 \cdot 1) \cdot \cos (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.7

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

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Step 2.1.2.7.1

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

$\frac{1}{4} \cdot \frac{1 + (0 - 1 \cdot 1) \cdot \cos (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 2.1.2.7.2

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

$\frac{1}{4} \cdot \frac{1 + (0 - 1 \cdot 1) \cdot \cos (0)}{lim_{x \to 0} x}$

Step 2.1.2.8

Simplify the answer.

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Step 2.1.2.8.1

Simplify each term.

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Step 2.1.2.8.1.1

Multiply $- 1$ by $1$ .

$\frac{1}{4} \cdot \frac{1 + (0 - 1) \cdot \cos (0)}{lim_{x \to 0} x}$

Step 2.1.2.8.1.2

Subtract $1$ from $0$ .

$\frac{1}{4} \cdot \frac{1 - 1 \cdot \cos (0)}{lim_{x \to 0} x}$

Step 2.1.2.8.1.3

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

$\frac{1}{4} \cdot \frac{1 - 1 \cdot 1}{lim_{x \to 0} x}$

Step 2.1.2.8.1.4

Multiply $- 1$ by $1$ .

$\frac{1}{4} \cdot \frac{1 - 1}{lim_{x \to 0} x}$

Step 2.1.2.8.2

Subtract $1$ from $1$ .

$\frac{1}{4} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

$\frac{1}{4} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{1}{4} \cdot \frac{0}{0}$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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 $f (x) = x - 1$ and $g (x) = \cos (x)$ .

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

Step 2.3.4.2

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

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

Step 2.3.4.3

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

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

Step 2.3.4.4

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

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

Step 2.3.4.5

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

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

Step 2.3.4.6

Add $1$ and $0$ .

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

Step 2.3.4.7

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Apply the distributive property.

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

Step 2.3.5.2

Combine terms.

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Step 2.3.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.5.2.2

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

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

Step 2.3.5.2.3

Add $0$ and $x (- \sin (x)) + \sin (x) + \cos (x)$ .

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

Step 2.3.5.3

Reorder terms.

$\frac{1}{4} lim_{x \to 0} \frac{- x \sin (x) + \sin (x) + \cos (x)}{\frac{d}{d x} [x]}$

Step 2.3.6

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

$\frac{1}{4} lim_{x \to 0} \frac{- x \sin (x) + \sin (x) + \cos (x)}{1}$

Step 2.4

Divide $- x \sin (x) + \sin (x) + \cos (x)$ by $1$ .

$\frac{1}{4} lim_{x \to 0} - x \sin (x) + \sin (x) + \cos (x)$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{1}{4} (- lim_{x \to 0} x \sin (x) + lim_{x \to 0} \sin (x) + lim_{x \to 0} \cos (x))$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

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Step 4.1

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

$\frac{1}{4} (0 \cdot \sin (lim_{x \to 0} x) + \sin (lim_{x \to 0} x) + \cos (lim_{x \to 0} x))$

Step 4.2

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

$\frac{1}{4} (0 \cdot \sin (0) + \sin (lim_{x \to 0} x) + \cos (lim_{x \to 0} x))$

Step 4.3

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

$\frac{1}{4} (0 \cdot \sin (0) + \sin (0) + \cos (lim_{x \to 0} x))$

Step 4.4

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

$\frac{1}{4} (0 \cdot \sin (0) + \sin (0) + \cos (0))$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$\frac{1}{4} (0 \cdot 0 + \sin (0) + \cos (0))$

Step 5.1.2

Multiply $0$ by $0$ .

$\frac{1}{4} (0 + \sin (0) + \cos (0))$

Step 5.1.3

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

$\frac{1}{4} (0 + 0 + \cos (0))$

Step 5.1.4

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

$\frac{1}{4} (0 + 0 + 1)$

Step 5.2

Add $0$ and $0$ .

$\frac{1}{4} (0 + 1)$

Step 5.3

Add $0$ and $1$ .

$\frac{1}{4} \cdot 1$

Step 5.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$