Question

Find the limit. use l'hospital's rule if appropriate. lim x→infinity x(tan 7/x

Answer 1

`\displaystyle\lim_(x\to\infty)x\tan\frac7x=7\lim_(x\to\infty)(\sin\frac7x)/(\frac7x\cos\frac7x)`

Replace
`y=\frac7x`. Then as
`x\to\infty`, you have
`y\to0^+`, so the limit is equivalent to


`\displaystyle7\lim_(y\to0^+)(\sin y)/(y\cos y)=7\left(\lim_(y\to0^+)\frac{\sin y}y\right)\left(\lim_(y\to0^+)\frac1{\cos y}\right)`

The first limit is well-known and has a value of 1. Meanwhile
`\cos y` is continuous at
`y=0` so the second limit evaluates to
`\frac1{\cos0}=1`.

This means the limit must be 7.

No L'Hopital's rule needed!