Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

$\frac{1 + \cos (2 π)}{1 - \sin (π)}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$\frac{1 + \cos (0)}{1 - \sin (π)}$

Step 10.1.2

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

$\frac{1 + 1}{1 - \sin (π)}$

Step 10.1.3

Add $1$ and $1$ .

$\frac{2}{1 - \sin (π)}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{2}{1 - \sin (0)}$

Step 10.2.2

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

$\frac{2}{1 - 0}$

Step 10.2.3

Multiply $- 1$ by $0$ .

$\frac{2}{1 + 0}$

Step 10.2.4

Add $1$ and $0$ .

$\frac{2}{1}$

Step 10.3

Divide $2$ by $1$ .

$2$