Sunday, 15 September 2013

limits - Why is limlimitsxtoinftyfrac1sqrtx2+2xx equal to 0?



So I made one exercise, which was limx+1x22xx I solved this one by:



limx+1x22xxx22x+xx22x+x



limx+x212x+x2x



limx+12x+12=1




The next exercise is limx1x2+2xx
I thought that I could solve this one in the same way by:



limx1x2+2xxx2+2x+xx2+2x+x



limxx21+2x+x2x



limx1+2x+12=1



But apparently, the answer is 0...




Why can't the second one be solved in the same way as the first?
And why is the answer 0?


Answer



When x, you don't have 0 as denominator, but +, because when x,x22x|x| and, as x<0, |x|=x, so you have limx1xx=limx12x=0.


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