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Lipossig

Answered question

2021-10-30

Compute the following using limits. Show all work using calculus

lim_{x \to \infty} x \sin (\frac{π}{x})

Bertha Stark

Skilled2021-10-31Added 96 answers

Given:

lim_{x \to \infty} x \sin (\frac{π}{x})

x \sin (\frac{π}{x}) = \frac{\sin \frac{π}{x}}{\frac{1}{x}}

= \frac{π}{π} \frac{\sin \frac{π}{x}}{\frac{1}{x}}

= π \frac{\sin \frac{π}{x}}{\frac{π}{x}}

θ = \frac{π}{x}

we have

lim_{x \to \infty} x \sin (\frac{π}{x}) = lim_{x \to \infty} π \sin (\frac{\frac{π}{x}}{\frac{π}{x}})

= π lim_{x \to \infty} \sin (\frac{\frac{π}{x}}{\frac{π}{x}})

= π lim_{θ \to 0} \frac{\sin θ}{θ}

= π

The line

y = π

is a horizontal asymptote of

y = x \sin (\frac{π}{x})

Jeffrey Jordon

Expert2022-06-28Added 2605 answers

Answer is given below (on video)

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