Calculus Examples

$lim_{x \to 0} \frac{5 \sin (2 x) - 12 x}{\sin (5 x)}$

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.

$\frac{lim_{x \to 0} 5 \sin (2 x) - 12 x}{lim_{x \to 0} \sin (5 x)}$

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 $0$ .

$\frac{lim_{x \to 0} 5 \sin (2 x) - lim_{x \to 0} 12 x}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.2

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

$\frac{5 lim_{x \to 0} \sin (2 x) - lim_{x \to 0} 12 x}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.3

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

$\frac{5 \sin (lim_{x \to 0} 2 x) - lim_{x \to 0} 12 x}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.4

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

$\frac{5 \sin (2 lim_{x \to 0} x) - lim_{x \to 0} 12 x}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.5

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 \sin (2 lim_{x \to 0} x) - 12 lim_{x \to 0} x}{lim_{x \to 0} \sin (5 x)}$

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 $x$ .

$\frac{5 \sin (2 \cdot 0) - 12 lim_{x \to 0} x}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.6.2

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

$\frac{5 \sin (2 \cdot 0) - 12 \cdot 0}{lim_{x \to 0} \sin (5 x)}$

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

Multiply $2$ by $0$ .

$\frac{5 \sin (0) - 12 \cdot 0}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.7.1.2

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

$\frac{5 \cdot 0 - 12 \cdot 0}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.7.1.3

Multiply $5$ by $0$ .

$\frac{0 - 12 \cdot 0}{lim_{x \to 0} \sin (5 x)}$

Step 1.1.2.7.1.4

Multiply $- 12$ by $0$ .

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

Step 1.1.2.7.2

Add $0$ and $0$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

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 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $5$ by $0$ .

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

Step 1.1.3.3.2

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

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 0} \frac{5 \sin (2 x) - 12 x}{\sin (5 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [5 \sin (2 x) - 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [5 \sin (2 x) - 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $5 \sin (2 x) - 12 x$ with respect to $x$ is $\frac{d}{d x} [5 \sin (2 x)] + \frac{d}{d x} [- 12 x]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [5 \sin (2 x)] + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [5 \sin (2 x)]$ .

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Step 1.3.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \sin (2 x)$ with respect to $x$ is $5 \frac{d}{d x} [\sin (2 x)]$ .

$lim_{x \to 0} \frac{5 \frac{d}{d x} [\sin (2 x)] + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.2

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

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Step 1.3.3.2.1

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

$lim_{x \to 0} \frac{5 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.2.2

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

$lim_{x \to 0} \frac{5 (\cos (u) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.2.3

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

$lim_{x \to 0} \frac{5 (\cos (2 x) \frac{d}{d x} [2 x]) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.3

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

$lim_{x \to 0} \frac{5 (\cos (2 x) (2 \frac{d}{d x} [x])) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.4

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

$lim_{x \to 0} \frac{5 (\cos (2 x) (2 \cdot 1)) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.5

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{5 (\cos (2 x) \cdot 2) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.6

Move $2$ to the left of $\cos (2 x)$ .

$lim_{x \to 0} \frac{5 (2 \cdot \cos (2 x)) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.3.7

Multiply $2$ by $5$ .

$lim_{x \to 0} \frac{10 \cos (2 x) + \frac{d}{d x} [- 12 x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 12 x]$ .

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Step 1.3.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{10 \cos (2 x) - 12 \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{10 \cos (2 x) - 12 \cdot 1}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.4.3

Multiply $- 12$ by $1$ .

$lim_{x \to 0} \frac{10 \cos (2 x) - 12}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.5

Reorder terms.

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\frac{d}{d x} [\sin (5 x)]}$

Step 1.3.6

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 1.3.6.1

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

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x]}$

Step 1.3.6.2

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

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\cos (u) \frac{d}{d x} [5 x]}$

Step 1.3.6.3

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

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\cos (5 x) \frac{d}{d x} [5 x]}$

Step 1.3.7

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

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\cos (5 x) (5 \frac{d}{d x} [x])}$

Step 1.3.8

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

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\cos (5 x) (5 \cdot 1)}$

Step 1.3.9

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\cos (5 x) \cdot 5}$

Step 1.3.10

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

$lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{5 \cos (5 x)}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{1}{5} lim_{x \to 0} \frac{- 12 + 10 \cos (2 x)}{\cos (5 x)}$

Step 2.2

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

$\frac{1}{5} \cdot \frac{lim_{x \to 0} - 12 + 10 \cos (2 x)}{lim_{x \to 0} \cos (5 x)}$

Step 2.3

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

$\frac{1}{5} \cdot \frac{- lim_{x \to 0} 12 + lim_{x \to 0} 10 \cos (2 x)}{lim_{x \to 0} \cos (5 x)}$

Step 2.4

Evaluate the limit of $12$ which is constant as $x$ approaches $0$ .

$\frac{1}{5} \cdot \frac{- 1 \cdot 12 + lim_{x \to 0} 10 \cos (2 x)}{lim_{x \to 0} \cos (5 x)}$

Step 2.5

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} \cdot \frac{- 12 + 10 lim_{x \to 0} \cos (2 x)}{lim_{x \to 0} \cos (5 x)}$

Step 2.6

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

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (lim_{x \to 0} 2 x)}{lim_{x \to 0} \cos (5 x)}$

Step 2.7

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

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (2 lim_{x \to 0} x)}{lim_{x \to 0} \cos (5 x)}$

Step 2.8

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

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (2 lim_{x \to 0} x)}{\cos (lim_{x \to 0} 5 x)}$

Step 2.9

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

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (2 lim_{x \to 0} x)}{\cos (5 lim_{x \to 0} x)}$

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 $x$ .

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (2 \cdot 0)}{\cos (5 lim_{x \to 0} x)}$

Step 3.2

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

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (2 \cdot 0)}{\cos (5 \cdot 0)}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $2$ by $0$ .

$\frac{1}{5} \cdot \frac{- 12 + 10 \cos (0)}{\cos (5 \cdot 0)}$

Step 4.1.2

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

$\frac{1}{5} \cdot \frac{- 12 + 10 \cdot 1}{\cos (5 \cdot 0)}$

Step 4.1.3

Multiply $10$ by $1$ .

$\frac{1}{5} \cdot \frac{- 12 + 10}{\cos (5 \cdot 0)}$

Step 4.1.4

Add $- 12$ and $10$ .

$\frac{1}{5} \cdot \frac{- 2}{\cos (5 \cdot 0)}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $5$ by $0$ .

$\frac{1}{5} \cdot \frac{- 2}{\cos (0)}$

Step 4.2.2

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

$\frac{1}{5} \cdot \frac{- 2}{1}$

Step 4.3

Divide $- 2$ by $1$ .

$\frac{1}{5} \cdot - 2$

Step 4.4

Combine $\frac{1}{5}$ and $- 2$ .

$\frac{- 2}{5}$

Step 4.5

Move the negative in front of the fraction.

$- \frac{2}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{5}$

Decimal Form:

$- 0.4$