Calculus Examples

$lim_{x \to 0} \frac{x \cos (3 x)}{\sin (5 x)}$

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.

$\frac{lim_{x \to 0} x \cos (3 x)}{lim_{x \to 0} \sin (5 x)}$

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 $0$ .

$\frac{lim_{x \to 0} x \cdot lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} \sin (5 x)}$

Step 1.2.2

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

$\frac{lim_{x \to 0} x \cdot \cos (lim_{x \to 0} 3 x)}{lim_{x \to 0} \sin (5 x)}$

Step 1.2.3

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

$\frac{lim_{x \to 0} x \cdot \cos (3 lim_{x \to 0} x)}{lim_{x \to 0} \sin (5 x)}$

Step 1.2.4

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

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Step 1.2.4.1

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

$\frac{0 \cdot \cos (3 lim_{x \to 0} x)}{lim_{x \to 0} \sin (5 x)}$

Step 1.2.4.2

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

$\frac{0 \cdot \cos (3 \cdot 0)}{lim_{x \to 0} \sin (5 x)}$

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Multiply $3$ by $0$ .

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

Step 1.2.5.2

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

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

Step 1.2.5.3

Multiply $0$ by $1$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.2

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

$\frac{0}{\sin (5 \cdot 0)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $5$ by $0$ .

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

Step 1.3.3.2

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

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{x \cos (3 x)}{\sin (5 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x \cos (3 x)]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x \cos (3 x)]}{\frac{d}{d x} [\sin (5 x)]}$

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

$lim_{x \to 0} \frac{x \frac{d}{d x} [\cos (3 x)] + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

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

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Step 3.3.1

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

$lim_{x \to 0} \frac{x (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [3 x]) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.3.2

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

$lim_{x \to 0} \frac{x (- \sin (u) \frac{d}{d x} [3 x]) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.3.3

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

$lim_{x \to 0} \frac{x (- \sin (3 x) \frac{d}{d x} [3 x]) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.4

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

$lim_{x \to 0} \frac{x (- \sin (3 x) (3 \frac{d}{d x} [x])) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.5

Multiply $3$ by $- 1$ .

$lim_{x \to 0} \frac{x (- 3 \sin (3 x) \frac{d}{d x} [x]) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.6

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

$lim_{x \to 0} \frac{x (- 3 \sin (3 x) \cdot 1) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.7

Multiply $- 3$ by $1$ .

$lim_{x \to 0} \frac{x (- 3 \sin (3 x)) + \cos (3 x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.8

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

$lim_{x \to 0} \frac{x (- 3 \sin (3 x)) + \cos (3 x) \cdot 1}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.9

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

$lim_{x \to 0} \frac{x (- 3 \sin (3 x)) + \cos (3 x)}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.10

Reorder terms.

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\frac{d}{d x} [\sin (5 x)]}$

Step 3.11

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 $g (x) = 5 x$ .

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Step 3.11.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x]}$

Step 3.11.2

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

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\cos (u) \frac{d}{d x} [5 x]}$

Step 3.11.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\cos (5 x) \frac{d}{d x} [5 x]}$

Step 3.12

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

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\cos (5 x) (5 \frac{d}{d x} [x])}$

Step 3.13

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

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\cos (5 x) (5 \cdot 1)}$

Step 3.14

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\cos (5 x) \cdot 5}$

Step 3.15

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

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{5 \cdot \cos (5 x)}$

Step 3.16

Multiply $5$ by $\cos (5 x)$ .

$lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{5 \cos (5 x)}$

Step 4

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

$\frac{1}{5} lim_{x \to 0} \frac{- 3 x \sin (3 x) + \cos (3 x)}{\cos (5 x)}$

Step 5

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

$\frac{1}{5} \cdot \frac{lim_{x \to 0} - 3 x \sin (3 x) + \cos (3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 6

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

$\frac{1}{5} \cdot \frac{- lim_{x \to 0} 3 x \sin (3 x) + lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 7

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \sin (3 x) + lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 8

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 9

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} 3 x) + lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 10

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 11

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + \cos (lim_{x \to 0} 3 x)}{lim_{x \to 0} \cos (5 x)}$

Step 12

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + \cos (3 lim_{x \to 0} x)}{lim_{x \to 0} \cos (5 x)}$

Step 13

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + \cos (3 lim_{x \to 0} x)}{\cos (lim_{x \to 0} 5 x)}$

Step 14

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

$\frac{1}{5} \cdot \frac{- 3 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + \cos (3 lim_{x \to 0} x)}{\cos (5 lim_{x \to 0} x)}$

Step 15

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

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Step 15.1

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

$\frac{1}{5} \cdot \frac{- 3 \cdot 0 \cdot \sin (3 lim_{x \to 0} x) + \cos (3 lim_{x \to 0} x)}{\cos (5 lim_{x \to 0} x)}$

Step 15.2

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

$\frac{1}{5} \cdot \frac{- 3 \cdot 0 \cdot \sin (3 \cdot 0) + \cos (3 lim_{x \to 0} x)}{\cos (5 lim_{x \to 0} x)}$

Step 15.3

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

$\frac{1}{5} \cdot \frac{- 3 \cdot 0 \cdot \sin (3 \cdot 0) + \cos (3 \cdot 0)}{\cos (5 lim_{x \to 0} x)}$

Step 15.4

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

$\frac{1}{5} \cdot \frac{- 3 \cdot 0 \cdot \sin (3 \cdot 0) + \cos (3 \cdot 0)}{\cos (5 \cdot 0)}$

Step 16

Simplify the answer.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Multiply $- 3$ by $0$ .

$\frac{1}{5} \cdot \frac{0 \cdot \sin (3 \cdot 0) + \cos (3 \cdot 0)}{\cos (5 \cdot 0)}$

Step 16.1.2

Multiply $3$ by $0$ .

$\frac{1}{5} \cdot \frac{0 \cdot \sin (0) + \cos (3 \cdot 0)}{\cos (5 \cdot 0)}$

Step 16.1.3

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

$\frac{1}{5} \cdot \frac{0 \cdot 0 + \cos (3 \cdot 0)}{\cos (5 \cdot 0)}$

Step 16.1.4

Multiply $0$ by $0$ .

$\frac{1}{5} \cdot \frac{0 + \cos (3 \cdot 0)}{\cos (5 \cdot 0)}$

Step 16.1.5

Multiply $3$ by $0$ .

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

Step 16.1.6

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

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

Step 16.1.7

Add $0$ and $1$ .

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

Step 16.2

Simplify the denominator.

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Step 16.2.1

Multiply $5$ by $0$ .

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

Step 16.2.2

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

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

Step 16.3

Cancel the common factor of $1$ .

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Step 16.3.1

Cancel the common factor.

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

Step 16.3.2

Rewrite the expression.

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

Step 16.4

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5}$