Question #041b9

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Answer 1 · 2017-05-14T15:34:46

1

$2^(1/x)/(1+2^(1/x)) = 2^u/(1+2^u)$ where $u$ approaches to $+oo$

$2^u/(1+2^u) = 1/(1/(2^u)+1)$

$u -> +oo <=> 2^u -> +oo$

then the limit will be $1/(0+1) = 1$

Answer 2 · 2017-05-14T15:42:41

$1.$

As $x to 0+, 1/x to +oo.$

Recall that, $"as "x to oo+, a^x to oo+," where "a gt 1.$

$:. 2^(1/x) to oo+," & so, "(1+2^(1/x)) to oo+," giving, "$

$lim_(x to 0+) 1/(1+2^(1/x))=0................(ast).$

$"Now, "2^(1/x)/(1+2^(1/x))={(1+2^(1/x))-1}/(1+2^(1/x))$

$=(1+2^(1/x))/(1+2^(1/x))-1/(1+2^(1/x))$

$=1-1/(1+2^(1/x)).$

$:.," by "(ast)," The Reqd. Lim.="1-0=1.$