note: x approaches 0 on the limit, didn't know how to format the code
I was solving this problem on my textbook:
On my first method, I replaced cos(x) with 1-sin^2(x/2) and after some modifications, the result turned out to be 3/2 (which was also the result on the textbook).
On the other hand, I tried this solution:
From the limit laws we get:
we know that limcos(x) as x approaches 0 is 1, so we end up:
lim((e^(x^2)-1)/x^2), which is equal to 1. Could anyone explain where I'm wrong on my second method?
You cannot use arithmetic of limits if you don't know that ALL the limits involved exist finitely
In the present case, since the limit in the DENOMINATOR equals zero...!
And after this you allow yourself to calculate ONLY the limit of the cosine but NOT the other ones....wrong.
Perhaps the simplest way is using L'Hospital's rule: