can someone teach me how can I solve this limit without using the L'Hopital's Rule?
limx→0(2+x22−x2)1x2
Thanks in advance.
Answer
2+x22−x2=1+2x22−x2=1+x21−x22
and y=x21−x22→0, as x→0. Hence
(1+x21−x22)1−x22x2=(1+y)1/y→e,
as x→0.
Finally
(2+x22−x2)1/x2=((1+x21−x22)1−x22x2)11−x22→e,
since, if f(x)→e and g(x)→1, then f(x)g(x)→e.
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