find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using…

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.\n\nlim x→∞ x sin(9π/x)\n\nresources\nread it watch it

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.\n\nlim x→∞ x sin(9π/x)\n\nresources\nread it watch it

Answer

Explanation:

Step1: Substitute ( t=\frac{1}{x} )

When ( x\to\infty ), then ( t\to0 ). So ( \lim_{x\to\infty}x\sin\left(\frac{9\pi}{x}\right)=\lim_{t\to0}\frac{\sin(9\pi t)}{t} )

Step2: Use the limit formula ( \lim_{u\to0}\frac{\sin u}{u} = 1 )

Let ( u = 9\pi t ). As ( t\to0 ), ( u\to0 ). Then ( \lim_{t\to0}\frac{\sin(9\pi t)}{t}=\lim_{t\to0}\frac{\sin(9\pi t)}{t}\times\frac{9\pi}{9\pi}=9\pi\lim_{t\to0}\frac{\sin(9\pi t)}{9\pi t} ) Since ( \lim_{u\to0}\frac{\sin u}{u} = 1 ), here ( u = 9\pi t ), so ( 9\pi\lim_{t\to0}\frac{\sin(9\pi t)}{9\pi t}=9\pi\times1 )

Answer:

( 9\pi )