Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\infty$ .

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

Step 1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

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

Step 1.3

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

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

Step 2

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

$\infty \cdot \cos (0)$

Step 3

Simplify the answer.

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Step 3.1

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

$\infty \cdot 1$

Step 3.2

Multiply $\infty$ by $1$ .

$\infty$