Calculus Examples

$lim_{x \to - 8} \cos (\frac{3}{x} + 3 π)$

Step 1

Evaluate the limit.

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Step 1.1

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

$\cos (lim_{x \to - 8} \frac{3}{x} + 3 π)$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\cos (lim_{x \to - 8} \frac{3}{x} + lim_{x \to - 8} 3 π)$

Step 1.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\cos (3 lim_{x \to - 8} \frac{1}{x} + lim_{x \to - 8} 3 π)$

Step 1.4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$\cos (3 \frac{lim_{x \to - 8} 1}{lim_{x \to - 8} x} + lim_{x \to - 8} 3 π)$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$\cos (3 \frac{1}{lim_{x \to - 8} x} + lim_{x \to - 8} 3 π)$

Step 1.6

Evaluate the limit of $3 π$ which is constant as $x$ approaches $- 8$ .

$\cos (3 \frac{1}{lim_{x \to - 8} x} + 3 π)$

Step 2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\cos (3 (\frac{1}{- 8}) + 3 π)$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Move the negative in front of the fraction.

$\cos (3 (- \frac{1}{8}) + 3 π)$

Step 3.1.2

Multiply $3 (- \frac{1}{8})$ .

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Step 3.1.2.1

Multiply $- 1$ by $3$ .

$\cos (- 3 (\frac{1}{8}) + 3 π)$

Step 3.1.2.2

Combine $- 3$ and $\frac{1}{8}$ .

$\cos (\frac{- 3}{8} + 3 π)$

Step 3.1.3

Move the negative in front of the fraction.

$\cos (- \frac{3}{8} + 3 π)$

Step 3.2

Evaluate $\cos (- \frac{3}{8} + 3 π)$ .

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