Calculus Examples

$lim_{x \to 0} \sin (x + 2 π) + \cos (x)$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} \sin (x + 2 π) + lim_{x \to 0} \cos (x)$

Step 2

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

$\sin (lim_{x \to 0} x + 2 π) + lim_{x \to 0} \cos (x)$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\sin (lim_{x \to 0} x + lim_{x \to 0} 2 π) + lim_{x \to 0} \cos (x)$

Step 4

Evaluate the limit of $2 π$ which is constant as $x$ approaches $0$ .

$\sin (lim_{x \to 0} x + 2 π) + lim_{x \to 0} \cos (x)$

Step 5

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

$\sin (lim_{x \to 0} x + 2 π) + \cos (lim_{x \to 0} x)$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sin (0 + 2 π) + \cos (lim_{x \to 0} x)$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sin (0 + 2 π) + \cos (0)$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Add $0$ and $2 π$ .

$\sin (2 π) + \cos (0)$

Step 7.1.2

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

$\sin (0) + \cos (0)$

Step 7.1.3

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

$0 + \cos (0)$

Step 7.1.4

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

$0 + 1$

Step 7.2

Add $0$ and $1$ .

$1$