How do to solve:Lim  (tg 5x)/(4x) whit x-> 0Thanks!must be resolved based on the remarkable limits

Expert Answers
sciencesolve eNotes educator| Certified Educator

Use the remarkable limit `lim_(u(x)->0)``(tan u(x))/u(x) = 1`

You should create this remarkable limit in your function, therefore you should divide and multiply by the argument of tangent function.

`lim_(x-gt0) ((tan 5x)/(5x))*(5x/4x) = lim_(x-gt0)((tan 5x)/(5x))*lim_(x-gt0)(5x)/(4x)`

`lim_(x-gt0)((tan 5x)/(4x))= 1*lim_(x-gt0)((5x)/(4x)) = 5/4`

Therefore, the limit of the fraction`((tan 5x)/(4x))`  is 5/4.