Calculus Examples

$lim_{x \to 0} \frac{1 - \cos (m x)}{1 - \cos (n x)}$

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.

$\frac{lim_{x \to 0} 1 - \cos (m x)}{lim_{x \to 0} 1 - \cos (n x)}$

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 $0$ .

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (m x)}{lim_{x \to 0} 1 - \cos (n x)}$

Step 1.1.2.1.2

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

$\frac{1 - lim_{x \to 0} \cos (m x)}{lim_{x \to 0} 1 - \cos (n x)}$

Step 1.1.2.1.3

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

$\frac{1 - \cos (lim_{x \to 0} m x)}{lim_{x \to 0} 1 - \cos (n x)}$

Step 1.1.2.1.4

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

$\frac{1 - \cos (m lim_{x \to 0} x)}{lim_{x \to 0} 1 - \cos (n x)}$

Step 1.1.2.2

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

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

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

Multiply $m$ by $0$ .

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

Step 1.1.2.3.1.2

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

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

Step 1.1.2.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

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

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

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

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

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

Step 1.1.3.1.4

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $n$ by $0$ .

$\frac{0}{1 - \cos (0)}$

Step 1.1.3.3.1.2

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

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

Step 1.1.3.3.1.3

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

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

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Step 1.3.4.2.1

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

$lim_{x \to 0} \frac{0 - (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [m x])}{\frac{d}{d x} [1 - \cos (n x)]}$

Step 1.3.4.2.2

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

$lim_{x \to 0} \frac{0 - (- \sin (u) \frac{d}{d x} [m x])}{\frac{d}{d x} [1 - \cos (n x)]}$

Step 1.3.4.2.3

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

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

Step 1.3.4.3

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

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

Step 1.3.4.4

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

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

Step 1.3.4.5

Multiply $m$ by $1$ .

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

Step 1.3.4.6

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.7

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

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $0$ and $\sin (m x) m$ .

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

Step 1.3.5.2

Reorder the factors of $\sin (m x) m$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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Step 1.3.8.1

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

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

Step 1.3.8.2

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

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Step 1.3.8.2.1

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

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

Step 1.3.8.2.2

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

$lim_{x \to 0} \frac{m \sin (m x)}{0 - (- \sin (u) \frac{d}{d x} [n x])}$

Step 1.3.8.2.3

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

$lim_{x \to 0} \frac{m \sin (m x)}{0 - (- \sin (n x) \frac{d}{d x} [n x])}$

Step 1.3.8.3

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

$lim_{x \to 0} \frac{m \sin (m x)}{0 - (- \sin (n x) (n \frac{d}{d x} [x]))}$

Step 1.3.8.4

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

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

Step 1.3.8.5

Multiply $n$ by $1$ .

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

Step 1.3.8.6

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{m \sin (m x)}{0 + 1 (\sin (n x) n)}$

Step 1.3.8.7

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

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

Step 1.3.9

Simplify.

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Step 1.3.9.1

Add $0$ and $\sin (n x) n$ .

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

Step 1.3.9.2

Reorder the factors of $\sin (n x) n$ .

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

Step 2

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

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

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.

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Multiply $m$ by $0$ .

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

Step 3.1.2.3.2

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

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

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

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

Step 3.1.3.1.2

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Multiply $n$ by $0$ .

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

Step 3.1.3.3.2

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

$\frac{m}{n} \cdot \frac{0}{0}$

Step 3.1.3.3.3

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

Undefined

$\frac{m}{n} \cdot \frac{0}{0}$

Step 3.1.3.4

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

Undefined

$\frac{m}{n} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{m}{n} \cdot \frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sin (m x)}{\sin (n x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (m x)]}{\frac{d}{d x} [\sin (n x)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{m}{n} lim_{x \to 0} \frac{\frac{d}{d x} [\sin (m x)]}{\frac{d}{d x} [\sin (n x)]}$

Step 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 $g (x) = m x$ .

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Step 3.3.2.1

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

$\frac{m}{n} lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [m x]}{\frac{d}{d x} [\sin (n x)]}$

Step 3.3.2.2

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

$\frac{m}{n} lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [m x]}{\frac{d}{d x} [\sin (n x)]}$

Step 3.3.2.3

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

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

Step 3.3.3

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

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

Step 3.3.4

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

$\frac{m}{n} lim_{x \to 0} \frac{\cos (m x) m \cdot 1}{\frac{d}{d x} [\sin (n x)]}$

Step 3.3.5

Multiply $m$ by $1$ .

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

Step 3.3.6

Reorder the factors of $\cos (m x) m$ .

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

Step 3.3.7

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 $g (x) = n x$ .

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Step 3.3.7.1

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

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [n x]}$

Step 3.3.7.2

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

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{\cos (u) \frac{d}{d x} [n x]}$

Step 3.3.7.3

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

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{\cos (n x) \frac{d}{d x} [n x]}$

Step 3.3.8

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

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{\cos (n x) n \frac{d}{d x} [x]}$

Step 3.3.9

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

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{\cos (n x) n \cdot 1}$

Step 3.3.10

Multiply $n$ by $1$ .

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{\cos (n x) n}$

Step 3.3.11

Reorder the factors of $\cos (n x) n$ .

$\frac{m}{n} lim_{x \to 0} \frac{m \cos (m x)}{n \cos (n x)}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{m}{n} \cdot \frac{m}{n} lim_{x \to 0} \frac{\cos (m x)}{\cos (n x)}$

Step 4.2

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

$\frac{m}{n} \cdot \frac{m}{n} \frac{lim_{x \to 0} \cos (m x)}{lim_{x \to 0} \cos (n x)}$

Step 4.3

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

$\frac{m}{n} \cdot \frac{m}{n} \frac{\cos (lim_{x \to 0} m x)}{lim_{x \to 0} \cos (n x)}$

Step 4.4

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

$\frac{m}{n} \cdot \frac{m}{n} \frac{\cos (m lim_{x \to 0} x)}{lim_{x \to 0} \cos (n x)}$

Step 4.5

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

$\frac{m}{n} \cdot \frac{m}{n} \frac{\cos (m lim_{x \to 0} x)}{\cos (lim_{x \to 0} n x)}$

Step 4.6

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

$\frac{m}{n} \cdot \frac{m}{n} \frac{\cos (m lim_{x \to 0} x)}{\cos (n lim_{x \to 0} x)}$

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 $x$ .

$\frac{m}{n} \cdot \frac{m}{n} \frac{\cos (m \cdot 0)}{\cos (n lim_{x \to 0} x)}$

Step 5.2

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

$\frac{m}{n} \cdot \frac{m}{n} \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6

Simplify the answer.

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Step 6.1

Multiply $\frac{m}{n} \cdot \frac{m}{n}$ .

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Step 6.1.1

Multiply $\frac{m}{n}$ by $\frac{m}{n}$ .

$\frac{m \cdot m}{n \cdot n} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.2

Raise $m$ to the power of $1$ .

$\frac{m^{1} m}{n \cdot n} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.3

Raise $m$ to the power of $1$ .

$\frac{m^{1} m^{1}}{n \cdot n} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.4

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

$\frac{m^{1 + 1}}{n \cdot n} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.5

Add $1$ and $1$ .

$\frac{m^{2}}{n \cdot n} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.6

Raise $n$ to the power of $1$ .

$\frac{m^{2}}{n^{1} n} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.7

Raise $n$ to the power of $1$ .

$\frac{m^{2}}{n^{1} n^{1}} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.8

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

$\frac{m^{2}}{n^{1 + 1}} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.1.9

Add $1$ and $1$ .

$\frac{m^{2}}{n^{2}} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.2

Combine.

$\frac{m^{2} \cos (m \cdot 0)}{n^{2} \cos (n \cdot 0)}$

Step 6.3

Separate fractions.

$\frac{m^{2}}{n^{2}} \cdot \frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$

Step 6.4

Rewrite $\frac{\cos (m \cdot 0)}{\cos (n \cdot 0)}$ as a product.

$\frac{m^{2}}{n^{2}} (\cos (m \cdot 0) \frac{1}{\cos (n \cdot 0)})$

Step 6.5

Write $\cos (m \cdot 0)$ as a fraction with denominator $1$ .

$\frac{m^{2}}{n^{2}} (\frac{\cos (m \cdot 0)}{1} \cdot \frac{1}{\cos (n \cdot 0)})$

Step 6.6

Simplify.

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Step 6.6.1

Divide $\cos (m \cdot 0)$ by $1$ .

$\frac{m^{2}}{n^{2}} (\cos (m \cdot 0) \frac{1}{\cos (n \cdot 0)})$

Step 6.6.2

Convert from $\frac{1}{\cos (n \cdot 0)}$ to $\sec (n \cdot 0)$ .

$\frac{m^{2}}{n^{2}} (\cos (m \cdot 0) \sec (n \cdot 0))$

Step 6.7

Multiply $m$ by $0$ .

$\frac{m^{2}}{n^{2}} (\cos (0) \sec (n \cdot 0))$

Step 6.8

Multiply $n$ by $0$ .

$\frac{m^{2}}{n^{2}} (\cos (0) \sec (0))$

Step 6.9

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

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Step 6.9.1

Reorder $\cos (0)$ and $\sec (0)$ .

$\frac{m^{2}}{n^{2}} (\sec (0) \cos (0))$

Step 6.9.2

Rewrite $\frac{m^{2}}{n^{2}} (\cos (0) \sec (0))$ in terms of sines and cosines.

$\frac{m^{2}}{n^{2}} (\frac{1}{\cos (0)} \cos (0))$

Step 6.9.3

Cancel the common factors.

$\frac{m^{2}}{n^{2}} \cdot 1$

Step 6.10

Multiply $\frac{m^{2}}{n^{2}}$ by $1$ .

$\frac{m^{2}}{n^{2}}$