Question

`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?

Answer 1

`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`