Expert: Ahmed Salami - 9/12/2010

Question
lim tan(5x)/ cos(x-pi/2)   
x→0

Answer
Hi Emma,
Directly substituting the limits result in the indeterminate form 0/0. We can therefore employ the L'Hopital's rule. It simply states that if the lim f(x)/g(x) results in the indeterminate form, then the limit can be evaluated as lim f'(x)/g'(x).
Now,
cos(x - π/2) = cos(π/2 - x) = sinx
So,
lim x→0  [tan(5x) / sinx]   =   lim x→0 [5sec²(5x) / cosx]   =   5/1 = 5

Regards