Find limx→∞(e((1+1x)x))x.
limx→∞(e((1+1x)x))x=limx→∞(ee)x=limx→∞1x=∞
Why is this incorrect?
Answer
Take the logarithm,
log(e(1+1x)x)x=xloge(1+1x)x=x(1−xlog(1+1x))=x(1−x(1x−12x2+O(x−3)))=x(1−1+12x+O(x−2))=12+O(x−1).
And hence
limx→∞(e(1+1x)x)x=e1/2.
Find limx→∞(e((1+1x)x))x.
limx→∞(e((1+1x)x))x=limx→∞(ee)x=limx→∞1x=∞
Why is this incorrect?
Answer
Take the logarithm,
log(e(1+1x)x)x=xloge(1+1x)x=x(1−xlog(1+1x))=x(1−x(1x−12x2+O(x−3)))=x(1−1+12x+O(x−2))=12+O(x−1).
And hence
limx→∞(e(1+1x)x)x=e1/2.
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