Calculus Examples

$\frac{lim_{x \to 3 π} \cos (x) + 1}{x - 3 π}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3 π$ .

$\frac{lim_{x \to 3 π} \cos (x) + lim_{x \to 3 π} 1}{x - 3 π}$

Step 1.2

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

$\frac{\cos (lim_{x \to 3 π} x) + lim_{x \to 3 π} 1}{x - 3 π}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $3 π$ .

$\frac{\cos (lim_{x \to 3 π} x) + 1}{x - 3 π}$

Step 2

Evaluate the limit of $x$ by plugging in $3 π$ for $x$ .

$\frac{\cos (3 π) + 1}{x - 3 π}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{\cos (π) + 1}{x - 3 π}$

Step 3.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{- \cos (0) + 1}{x - 3 π}$

Step 3.1.3

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

$\frac{- 1 \cdot 1 + 1}{x - 3 π}$

Step 3.1.4

Multiply $- 1$ by $1$ .

$\frac{- 1 + 1}{x - 3 π}$

Step 3.1.5

Add $- 1$ and $1$ .

$\frac{0}{x - 3 π}$

Step 3.2

Divide $0$ by $x - 3 π$ .

$0$