Calculus Examples

$lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x \cos (x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{π}{\frac{}{}} \cdot 2$ .

$lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x \cdot lim_{x \to \frac{π}{\frac{}{}} \cdot 2} \cos (x)$

Step 2

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

$lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3

Evaluate the limits by plugging in $\frac{π}{\frac{}{}} \cdot 2$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $\frac{π}{\frac{}{}} \cdot 2$ for $x$ .

$\frac{π}{\frac{}{}} \cdot 2 \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.1

Reduce the expression $\frac{}{}$ by cancelling the common factors.

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Step 3.2.1.1

Cancel the common factor.

$\frac{π}{\frac{}{}} \cdot 2 \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3.2.1.2

Rewrite the expression.

$\frac{π}{\frac{1}{1}} \cdot 2 \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3.2.2

Rewrite the expression.

$\frac{π}{1} \cdot 2 \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3.3

Divide $π$ by $1$ .

$π \cdot 2 \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3.4

Move $2$ to the left of $π$ .

$2 π \cdot \cos (lim_{x \to \frac{π}{\frac{}{}} \cdot 2} x)$

Step 3.5

Evaluate the limit of $x$ by plugging in $\frac{π}{\frac{}{}} \cdot 2$ for $x$ .

$2 π \cdot \cos (\frac{π}{\frac{}{}} \cdot 2)$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Reduce the expression $\frac{}{}$ by cancelling the common factors.

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Step 3.6.1.1

Cancel the common factor.

$2 π \cdot \cos (\frac{π}{\frac{}{}} \cdot 2)$

Step 3.6.1.2

Rewrite the expression.

$2 π \cdot \cos (\frac{π}{\frac{1}{1}} \cdot 2)$

Step 3.6.2

Rewrite the expression.

$2 π \cdot \cos (\frac{π}{1} \cdot 2)$

Step 3.7

Divide $π$ by $1$ .

$2 π \cdot \cos (π \cdot 2)$

Step 3.8

Move $2$ to the left of $π$ .

$2 π \cdot \cos (2 π)$

Step 4

Simplify the answer.

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Step 4.1

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

$2 π \cdot \cos (0)$

Step 4.2

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

$2 π \cdot 1$

Step 4.3

Multiply $2$ by $1$ .

$2 π$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 π$

Decimal Form:

$6.28318530 \dots$