SOLUTION: Please help me by simlifying the following logarithum function. log(log(1-x^2)/log(log(cosx). In fact I need to simplify it to find the limit of the above function when xtenda t

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Please help me by simlifying the following logarithum function. log(log(1-x^2)/log(log(cosx). In fact I need to simplify it to find the limit of the above function when xtenda t      Log On


   

Question 889953: Please help me by simlifying the following logarithum function.
log(log(1-x^2)/log(log(cosx). In fact I need to simplify it to find the limit of the above function when xtenda to Zero.
Regards,
S B Roy
Found 2 solutions by rothauserc, Edwin McCravy:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
as x approaches 0, we have
log(log(1-x^2)/log(log(cosx) = log(log(1)/log(log(cos0)
now the log 1 = 0 and cos(0) = 1
we now have
log(0) / log(1) = -infinity / 0
note that division by 0 is undefined, therefore this limit is undefined

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor's answer is wrong.

You don't need to simplify it.

Use the theorem that if f is continuous in a neighborhood of a,
and is not constant on any subinterval of that interval
and lim(x/y) as x->a exists,

lim [f(x)/f(y)] = lim(x/y)
x->a              x->a

then since cos(0) = 1,
                                               _                 _
lim  log[log(1-x^2)]      lim   1-x^2     lim |    1         x^2  |
     ----------------- =       ------- =      | ------- - --------| = 1-0 = 1
x->0  log[log(cos(x)]     x->0  cos(x)    x->0|_ cos(x)    cos(x)_|

Edwin