Calculus Examples

$lim_{x \to π} \frac{12 \cos (12 x)}{4 \cos (4 x)}$

Step 1

Cancel the common factor of $12$ and $4$ .

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Step 1.1

Factor $4$ out of $12 \cos (12 x)$ .

$lim_{x \to π} \frac{4 (3 \cos (12 x))}{4 \cos (4 x)}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $4$ out of $4 \cos (4 x)$ .

$lim_{x \to π} \frac{4 (3 \cos (12 x))}{4 (\cos (4 x))}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to π} \frac{4 (3 \cos (12 x))}{4 \cos (4 x)}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to π} \frac{3 \cos (12 x)}{\cos (4 x)}$

Step 2

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

$3 lim_{x \to π} \frac{\cos (12 x)}{\cos (4 x)}$

Step 3

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

$3 \frac{lim_{x \to π} \cos (12 x)}{lim_{x \to π} \cos (4 x)}$

Step 4

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

$3 \frac{\cos (lim_{x \to π} 12 x)}{lim_{x \to π} \cos (4 x)}$

Step 5

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

$3 \frac{\cos (12 lim_{x \to π} x)}{lim_{x \to π} \cos (4 x)}$

Step 6

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

$3 \frac{\cos (12 lim_{x \to π} x)}{\cos (lim_{x \to π} 4 x)}$

Step 7

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

$3 \frac{\cos (12 lim_{x \to π} x)}{\cos (4 lim_{x \to π} x)}$

Step 8

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

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Step 8.1

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

$3 \frac{\cos (12 π)}{\cos (4 lim_{x \to π} x)}$

Step 8.2

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

$3 \frac{\cos (12 π)}{\cos (4 π)}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$3 \frac{\cos (0)}{\cos (4 π)}$

Step 9.1.2

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

$3 \frac{1}{\cos (4 π)}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$3 \frac{1}{\cos (0)}$

Step 9.2.2

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

$3 (\frac{1}{1})$

Step 9.3

Cancel the common factor of $1$ .

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Step 9.3.1

Cancel the common factor.

$3 (\frac{1}{1})$

Step 9.3.2

Rewrite the expression.

$3 \cdot 1$

Step 9.4

Multiply $3$ by $1$ .

$3$