Calculus Examples

$lim_{x \to \infty} 2 \cos (\frac{π}{x})$

Step 1

Evaluate the limit.

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Step 1.1

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

$2 lim_{x \to \infty} \cos (\frac{π}{x})$

Step 1.2

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

$2 \cos (lim_{x \to \infty} \frac{π}{x})$

Step 1.3

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

$2 \cos (π lim_{x \to \infty} \frac{1}{x})$

Step 2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$2 \cos (π \cdot 0)$

Step 3

Simplify the answer.

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Step 3.1

Multiply $π$ by $0$ .

$2 \cos (0)$

Step 3.2

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

$2 \cdot 1$

Step 3.3

Multiply $2$ by $1$ .

$2$