find lim(lnx)/(sin180x)?

Answer 1

I'm going to change 180x into pi(x), because they mean the same thing.

lim [ ln(x) / sin(pi(x))
x -> 0+

Use L'Hospital's rule.

lim [ (1/x) / (pi)cos(pi*x) ]
x -> 0+

Change this complex fraction into a simple fraction.

lim [ 1/[x * pi * cos(pi*x) ]
x -> 0+

This is now in the form [1/0], but because this is one-sided, this is either going to approach infinity or negative infinity. We can easily see that it approaches infinity, knowing the behaviour of the function 1/x from the right.

Answer 2

I guess that you are looking for the limit as x goes to 0, from above. Right?

The limit is negative infinity.

The numerator is going to negative infinity, and the denominator is going to 0. As x goes to 0, the numerator is getting more and more negative. We are dividing by a number that is positive, but getting smaller and smaller. The division just makes the ratio go to negative infinity even faster than the numerator does.

L'Hospital's theorem does not apply here. It only applies when numerator and denominator are *both* going to infinity, or when they are *both* going to zero. In this problem, the numerator goes to -infinity, but the denominator is going to zero. That case is not covered by L'Hospital's rule.

Answer 3

when x goes where??