Calculus Examples

$lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} \frac{\cos (x)}{1 - \sin (x)}$

Step 1

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

$\frac{lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} \cos (x)}{lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} 1 - \sin (x)}$

Step 2

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

$\frac{\cos (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}{lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} 1 - \sin (x)}$

Step 3

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

$\frac{\cos (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}{lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} 1 - lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} \sin (x)}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{\frac{}{}} \cdot 2$ .

$\frac{\cos (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}{1 - lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} \sin (x)}$

Step 5

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

$\frac{\cos (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6

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

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Step 6.1

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

$\frac{\cos (\frac{π}{\frac{}{}} \cdot 2)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{\cos (\frac{π}{\frac{}{}} \cdot 2)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{\cos (\frac{π}{\frac{1}{1}} \cdot 2)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6.2.2

Rewrite the expression.

$\frac{\cos (\frac{π}{1} \cdot 2)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6.3

Divide $π$ by $1$ .

$\frac{\cos (π \cdot 2)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6.4

Move $2$ to the left of $π$ .

$\frac{\cos (2 π)}{1 - \sin (lim_{x \to {(\frac{π}{\frac{}{}} \cdot 2)}^{+}} x)}$

Step 6.5

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

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

Step 6.6

Reduce the expression by cancelling the common factors.

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Step 6.6.1

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

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Step 6.6.1.1

Cancel the common factor.

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

Step 6.6.1.2

Rewrite the expression.

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

Step 6.6.2

Rewrite the expression.

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

Step 6.7

Divide $π$ by $1$ .

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

Step 6.8

Move $2$ to the left of $π$ .

$\frac{\cos (2 π)}{1 - \sin (2 π)}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{\cos (0)}{1 - \sin (2 π)}$

Step 7.1.2

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

$\frac{1}{1 - \sin (2 π)}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Multiply $- 1$ by $0$ .

$\frac{1}{1 + 0}$

Step 7.2.4

Add $1$ and $0$ .

$\frac{1}{1}$

Step 7.3

Cancel the common factor of $1$ .

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Step 7.3.1

Cancel the common factor.

$\frac{1}{1}$

Step 7.3.2

Rewrite the expression.

$1$