Calculus Examples

$lim_{x \to 4} c x + \cos (π x)$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$lim_{x \to 4} c x + lim_{x \to 4} \cos (π x)$

Step 2

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

$c lim_{x \to 4} x + lim_{x \to 4} \cos (π x)$

Step 3

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

$c lim_{x \to 4} x + \cos (lim_{x \to 4} π x)$

Step 4

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

$c lim_{x \to 4} x + \cos (π lim_{x \to 4} x)$

Step 5

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$c \cdot 4 + \cos (π lim_{x \to 4} x)$

Step 5.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$c \cdot 4 + \cos (π \cdot 4)$

Step 6

Simplify each term.

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Step 6.1

Move $4$ to the left of $c$ .

$4 \cdot c + \cos (π \cdot 4)$

Step 6.2

Move $4$ to the left of $π$ .

$4 c + \cos (4 \cdot π)$

Step 6.3

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

$4 c + \cos (0)$

Step 6.4

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

$4 c + 1$