Calculus Examples

$lim_{x \to 0} \frac{\sin (4 x) \sec (5 x)}{5 x}$

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.

$\frac{lim_{x \to 0} \sin (4 x) \sec (5 x)}{lim_{x \to 0} 5 x}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{lim_{x \to 0} \sin (4 x) \cdot lim_{x \to 0} \sec (5 x)}{lim_{x \to 0} 5 x}$

Step 1.2.2

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

$\frac{\sin (lim_{x \to 0} 4 x) \cdot lim_{x \to 0} \sec (5 x)}{lim_{x \to 0} 5 x}$

Step 1.2.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (4 lim_{x \to 0} x) \cdot lim_{x \to 0} \sec (5 x)}{lim_{x \to 0} 5 x}$

Step 1.2.4

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

$\frac{\sin (4 lim_{x \to 0} x) \cdot \sec (lim_{x \to 0} 5 x)}{lim_{x \to 0} 5 x}$

Step 1.2.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (4 lim_{x \to 0} x) \cdot \sec (5 lim_{x \to 0} x)}{lim_{x \to 0} 5 x}$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{\sin (4 \cdot 0) \cdot \sec (5 lim_{x \to 0} x)}{lim_{x \to 0} 5 x}$

Step 1.2.6.2

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

$\frac{\sin (4 \cdot 0) \cdot \sec (5 \cdot 0)}{lim_{x \to 0} 5 x}$

Step 1.2.7

Simplify the answer.

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Step 1.2.7.1

Multiply $4$ by $0$ .

$\frac{\sin (0) \cdot \sec (5 \cdot 0)}{lim_{x \to 0} 5 x}$

Step 1.2.7.2

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

$\frac{0 \cdot \sec (5 \cdot 0)}{lim_{x \to 0} 5 x}$

Step 1.2.7.3

Multiply $5$ by $0$ .

$\frac{0 \cdot \sec (0)}{lim_{x \to 0} 5 x}$

Step 1.2.7.4

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

$\frac{0 \cdot 1}{lim_{x \to 0} 5 x}$

Step 1.2.7.5

Multiply $0$ by $1$ .

$\frac{0}{lim_{x \to 0} 5 x}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{5 lim_{x \to 0} x}$

Step 1.3.2

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

$\frac{0}{5 \cdot 0}$

Step 1.3.3

Multiply $5$ by $0$ .

$\frac{0}{0}$

Step 1.3.4

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.

$lim_{x \to 0} \frac{\sin (4 x) \sec (5 x)}{5 x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (4 x) \sec (5 x)]}{\frac{d}{d x} [5 x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (4 x) \sec (5 x)]}{\frac{d}{d x} [5 x]}$

Step 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 $f (x) = \sin (4 x)$ and $g (x) = \sec (5 x)$ .

$lim_{x \to 0} \frac{\sin (4 x) \frac{d}{d x} [\sec (5 x)] + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 5 x$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$lim_{x \to 0} \frac{\sin (4 x) (\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [5 x]) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.3.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$lim_{x \to 0} \frac{\sin (4 x) (\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [5 x]) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$lim_{x \to 0} \frac{\sin (4 x) (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x]) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.4

Remove parentheses.

$lim_{x \to 0} \frac{\sin (4 x) (\sec (5 x) \tan (5 x)) \frac{d}{d x} [5 x] + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.5

Remove parentheses.

$lim_{x \to 0} \frac{\sin (4 x) \sec (5 x) \tan (5 x) \frac{d}{d x} [5 x] + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.6

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\sin (4 x) \sec (5 x) \tan (5 x) (5 \frac{d}{d x} [x]) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.7

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

$lim_{x \to 0} \frac{\sin (4 x) \sec (5 x) \tan (5 x) (5 \cdot 1) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.8

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{\sin (4 x) \sec (5 x) \tan (5 x) \cdot 5 + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.9

Move $5$ to the left of $\sin (4 x) \sec (5 x) \tan (5 x)$ .

$lim_{x \to 0} \frac{5 \cdot (\sin (4 x) \sec (5 x) \tan (5 x)) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.10

Remove parentheses.

$lim_{x \to 0} \frac{5 \cdot (\sin (4 x) \sec (5 x)) \tan (5 x) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.11

Remove parentheses.

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [5 x]}$

Step 3.12

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 $g (x) = 4 x$ .

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Step 3.12.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [4 x])}{\frac{d}{d x} [5 x]}$

Step 3.12.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) (\cos (u_{2}) \frac{d}{d x} [4 x])}{\frac{d}{d x} [5 x]}$

Step 3.12.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) (\cos (4 x) \frac{d}{d x} [4 x])}{\frac{d}{d x} [5 x]}$

Step 3.13

Remove parentheses.

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) \cos (4 x) \frac{d}{d x} [4 x]}{\frac{d}{d x} [5 x]}$

Step 3.14

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) \cos (4 x) (4 \frac{d}{d x} [x])}{\frac{d}{d x} [5 x]}$

Step 3.15

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

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) \cos (4 x) (4 \cdot 1)}{\frac{d}{d x} [5 x]}$

Step 3.16

Multiply $4$ by $1$ .

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + \sec (5 x) \cos (4 x) \cdot 4}{\frac{d}{d x} [5 x]}$

Step 3.17

Move $4$ to the left of $\sec (5 x) \cos (4 x)$ .

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + 4 \cdot (\sec (5 x) \cos (4 x))}{\frac{d}{d x} [5 x]}$

Step 3.18

Remove parentheses.

$lim_{x \to 0} \frac{5 \sin (4 x) \sec (5 x) \tan (5 x) + 4 \sec (5 x) \cos (4 x)}{\frac{d}{d x} [5 x]}$

Step 3.19

Reorder terms.

$lim_{x \to 0} \frac{5 \sec (5 x) \sin (4 x) \tan (5 x) + 4 \cos (4 x) \sec (5 x)}{\frac{d}{d x} [5 x]}$

Step 3.20

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 \sec (5 x) \sin (4 x) \tan (5 x) + 4 \cos (4 x) \sec (5 x)}{5 \frac{d}{d x} [x]}$

Step 3.21

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

$lim_{x \to 0} \frac{5 \sec (5 x) \sin (4 x) \tan (5 x) + 4 \cos (4 x) \sec (5 x)}{5 \cdot 1}$

Step 3.22

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{5 \sec (5 x) \sin (4 x) \tan (5 x) + 4 \cos (4 x) \sec (5 x)}{5}$

Step 4

Move the term $\frac{1}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} lim_{x \to 0} 5 \sec (5 x) \sin (4 x) \tan (5 x) + 4 \cos (4 x) \sec (5 x)$

Step 5

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

$\frac{1}{5} (lim_{x \to 0} 5 \sec (5 x) \sin (4 x) \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 lim_{x \to 0} \sec (5 x) \sin (4 x) \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 7

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

$\frac{1}{5} (5 lim_{x \to 0} \sec (5 x) \cdot lim_{x \to 0} \sin (4 x) \cdot lim_{x \to 0} \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 8

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

$\frac{1}{5} (5 \sec (lim_{x \to 0} 5 x) \cdot lim_{x \to 0} \sin (4 x) \cdot lim_{x \to 0} \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 9

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (4 x) \cdot lim_{x \to 0} \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 10

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

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 4 x) \cdot lim_{x \to 0} \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 11

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot lim_{x \to 0} \tan (5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 12

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

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (lim_{x \to 0} 5 x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 13

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + lim_{x \to 0} 4 \cos (4 x) \sec (5 x))$

Step 14

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos (4 x) \sec (5 x))$

Step 15

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

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos (4 x) \cdot lim_{x \to 0} \sec (5 x))$

Step 16

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

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 \cos (lim_{x \to 0} 4 x) \cdot lim_{x \to 0} \sec (5 x))$

Step 17

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 \cos (4 lim_{x \to 0} x) \cdot lim_{x \to 0} \sec (5 x))$

Step 18

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

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 \cos (4 lim_{x \to 0} x) \cdot \sec (lim_{x \to 0} 5 x))$

Step 19

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} (5 \sec (5 lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 \cos (4 lim_{x \to 0} x) \cdot \sec (5 lim_{x \to 0} x))$

Step 20

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

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Step 20.1

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

$\frac{1}{5} (5 \sec (5 \cdot 0) \cdot \sin (4 lim_{x \to 0} x) \cdot \tan (5 lim_{x \to 0} x) + 4 \cos (4 lim_{x \to 0} x) \cdot \sec (5 lim_{x \to 0} x))$

Step 20.2

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

$\frac{1}{5} (5 \sec (5 \cdot 0) \cdot \sin (4 \cdot 0) \cdot \tan (5 lim_{x \to 0} x) + 4 \cos (4 lim_{x \to 0} x) \cdot \sec (5 lim_{x \to 0} x))$

Step 20.3

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

$\frac{1}{5} (5 \sec (5 \cdot 0) \cdot \sin (4 \cdot 0) \cdot \tan (5 \cdot 0) + 4 \cos (4 lim_{x \to 0} x) \cdot \sec (5 lim_{x \to 0} x))$

Step 20.4

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

$\frac{1}{5} (5 \sec (5 \cdot 0) \cdot \sin (4 \cdot 0) \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 lim_{x \to 0} x))$

Step 20.5

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

$\frac{1}{5} (5 \sec (5 \cdot 0) \cdot \sin (4 \cdot 0) \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21

Simplify the answer.

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Step 21.1

Simplify each term.

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Step 21.1.1

Multiply $5$ by $0$ .

$\frac{1}{5} (5 \sec (0) \cdot \sin (4 \cdot 0) \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.2

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

$\frac{1}{5} (5 \cdot 1 \cdot \sin (4 \cdot 0) \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.3

Multiply $5$ by $1$ .

$\frac{1}{5} (5 \cdot \sin (4 \cdot 0) \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.4

Multiply $4$ by $0$ .

$\frac{1}{5} (5 \cdot \sin (0) \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.5

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

$\frac{1}{5} (5 \cdot 0 \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.6

Multiply $5$ by $0$ .

$\frac{1}{5} (0 \cdot \tan (5 \cdot 0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.7

Multiply $5$ by $0$ .

$\frac{1}{5} (0 \cdot \tan (0) + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.8

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

$\frac{1}{5} (0 \cdot 0 + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.9

Multiply $0$ by $0$ .

$\frac{1}{5} (0 + 4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0))$

Step 21.1.10

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 21.1.10.1

Add parentheses.

$\frac{1}{5} (0 + 4 (\cos (4 \cdot 0) \cdot \sec (5 \cdot 0)))$

Step 21.1.10.2

Reorder $\cos (4 \cdot 0)$ and $\sec (5 \cdot 0)$ .

$\frac{1}{5} (0 + 4 (\sec (5 \cdot 0) \cdot \cos (4 \cdot 0)))$

Step 21.1.10.3

Factor $5$ out of $4 \cdot 0$ .

$\frac{1}{5} (0 + 4 (\sec (5 \cdot 0) \cdot \cos (5 (4 \cdot 0))))$

Step 21.1.10.4

Multiply $4$ by $0$ .

$\frac{1}{5} (0 + 4 (\sec (5 \cdot 0) \cdot \cos (5 \cdot 0)))$

Step 21.1.10.5

Rewrite $4 \cos (4 \cdot 0) \cdot \sec (5 \cdot 0)$ in terms of sines and cosines.

$\frac{1}{5} (0 + 4 (\frac{1}{\cos (5 \cdot 0)} \cos (5 \cdot 0)))$

Step 21.1.10.6

Cancel the common factors.

$\frac{1}{5} (0 + 4 \cdot 1)$

Step 21.1.11

Multiply $4$ by $1$ .

$\frac{1}{5} (0 + 4)$

Step 21.2

Add $0$ and $4$ .

$\frac{1}{5} \cdot 4$

Step 21.3

Combine $\frac{1}{5}$ and $4$ .

$\frac{4}{5}$