#lim_(x to 0^+) (2-e^sqrt(x))^(1/x)# solve for x?

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Answer 1 · 2017-12-30T19:21:21

#Lim_(xrarr0^+)(2−e^sqrt(x))^(1/x)=0#

#Lim_(xrarr0^+)(2−e^sqrt(x))^(1/x)=1^oo#

Applying euler's identity: #e^(lnx)=x#

#Lim_(xrarr0^+)e^(ln(2−e^sqrt(x))^(1/x))=Lim_(xrarr0^+)e^(1/xln(2−e^sqrt(x)))#

e is a number, that means it can go in front of limit like this:

#e^(Lim_(xrarr0^+)1/xln(2−e^sqrt(x)))#

Let's find limit first:

#Lim_(xrarr0^+)(ln(2−e^sqrt(x)))/x=0/0#

Using L'hopitals rule:

#Lim_(xrarr0^+)(1/(2−e^sqrt(x))*(-e^(sqrtx))1/2x^(-1/2))/1#

#Lim_(xrarr0^+)(-e^(sqrtx))/(2(2−e^sqrt(x))x^(1/2))=-1/0^+=-oo#

#=>e^(-oo)=1/e^(oo)=0#