Question #041b9

2 Answers
May 14, 2017

1

Explanation:

2^(1/x)/(1+2^(1/x)) = 2^u/(1+2^u)21x1+21x=2u1+2u where uu approaches to +oo+

2^u/(1+2^u) = 1/(1/(2^u)+1)2u1+2u=112u+1

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

then the limit will be 1/(0+1) = 110+1=1

May 14, 2017

1.1.

Explanation:

As x to 0+, 1/x to +oo. x0+,1x+.

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