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

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

1 Answer
Dec 30, 2017

Lim_(xrarr0^+)(2−e^sqrt(x))^(1/x)=0

Explanation:

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