how do i do this?

(x)=xlnx-e^2^x+x^-2

This ca re-written like this:

x+e^(2^x) = 1/x^2+x log(x)

x = 0.458076114665514...

Limit:

limit( xlnx-(((e^2))^x)+(x^-2), x=0 ) = infinity 

{x=(-(((sqrt(3)*i)/2))-((1/2)))*((sqrt(4*xlnx^3-((12*(abs(e))^(2*x))*xlnx^2)+(12*(abs(e))^(4*x))*xlnx-(4*(abs(e))^(6*x))+27)/(2*3^((3/2))))-(((-(2*xlnx^3)+(6*(abs(e))^(2*x))*xlnx^2-((6*(abs(e))^(4*x))*xlnx)+2*(abs(e))^(6*x)-27)/54)))^((1/3))+(((((sqrt(3)*i)/2)-((1/2)))*((xlnx^2)-((2*(abs(e))^(2*x))*xlnx)+((abs(e))^(4*x))))/(9*((sqrt(4*xlnx^3-((12*(abs(e))^(2*x))*xlnx^2)+(12*(abs(e))^(4*x))*xlnx-(4*(abs(e))^(6*x))+27)/(2*3^((3/2))))-(((-(2*xlnx^3)+(6*(abs(e))^(2*x))*xlnx^2-((6*(abs(e))^(4*x))*xlnx)+2*(abs(e))^(6*x)-27)/54)))^((1/3))))-(((((abs(e))^(2*x))-xlnx)/3)), x=(((sqrt(3)*i)/2)-((1/2)))*((sqrt(4*xlnx^3-((12*(abs(e))^(2*x))*xlnx^2)+(12*(abs(e))^(4*x))*xlnx-(4*(abs(e))^(6*x))+27)/(2*3^((3/2))))-(((-(2*xlnx^3)+(6*(abs(e))^(2*x))*xlnx^2-((6*(abs(e))^(4*x))*xlnx)+2*(abs(e))^(6*x)-27)/54)))^((1/3))+(((-(((sqrt(3)*i)/2))-((1/2)))*((xlnx^2)-((2*(abs(e))^(2*x))*xlnx)+((abs(e))^(4*x))))/(9*((sqrt(4*xlnx^3-((12*(abs(e))^(2*x))*xlnx^2)+(12*(abs(e))^(4*x))*xlnx-(4*(abs(e))^(6*x))+27)/(2*3^((3/2))))-(((-(2*xlnx^3)+(6*(abs(e))^(2*x))*xlnx^2-((6*(abs(e))^(4*x))*xlnx)+2*(abs(e))^(6*x)-27)/54)))^((1/3))))-(((((abs(e))^(2*x))-xlnx)/3)), x=(((sqrt(4*xlnx^3-((12*(abs(e))^(2*x))*xlnx^2)+(12*(abs(e))^(4*x))*xlnx-(4*(abs(e))^(6*x))+27)/(2*3^((3/2))))-(((-(2*xlnx^3)+(6*(abs(e))^(2*x))*xlnx^2-((6*(abs(e))^(4*x))*xlnx)+2*(abs(e))^(6*x)-27)/54)))^((1/3)))+(((xlnx^2)-((2*(abs(e))^(2*x))*xlnx)+((abs(e))^(4*x)))/(9*((sqrt(4*xlnx^3-((12*(abs(e))^(2*x))*xlnx^2)+(12*(abs(e))^(4*x))*xlnx-(4*(abs(e))^(6*x))+27)/(2*3^((3/2))))-(((-(2*xlnx^3)+(6*(abs(e))^(2*x))*xlnx^2-((6*(abs(e))^(4*x))*xlnx)+2*(abs(e))^(6*x)-27)/54)))^((1/3))))-(((((abs(e))^(2*x))-xlnx)/3))}   is the answer

(x)=xlnx-e^2^x+x^-2

\(x=xlnx-(e^2)^x+x^{-2}\\\\ x=xlnx-e^{2x}+\frac{1}{x^{2}}\\\\ \)

Here is a graphical solution

https://www.desmos.com/calculator/g5xyac7btt

It shows that x=0.52, y=0.52 is the approximate solution.

One of the problems here is that I do not know what you mean by  e^2^x 

There is an order of operation definition but I can never remember what it is and it may not be what you mean anyway.

Do you mean

\((e^2)^x\;\;\;or\;\;\;e^{2^x}\)

they are different. :)

Here is a graphical solution to the other interpretation

https://www.desmos.com/calculator/jfh1wedmr4

The approximate solution is here x=0.458, y=0.458