Sorry, this is very hard to read.

"we can se that it will be <0": what will be < 0?

"we only will get positive x intercept": why are you talking about x-intercepts? And x-intercepts of what? The x-intercepts of the original function

(*)

do not arise in this problem at all.

"the x intercept when -(f(x)-L)<epsilon": you can't talk about the x-intercept "when" an equation holds. The concept of an x-intercept is only applicable to a function, not to an inequality. An inequality has solutions, which is a possibly infinite set of real numbers. Sometimes this set can be expressed using several inequalities of the form x > ... or x < ... .

"and get x intercept as 2.82 and 0.2528": the second value should be negative, but it is not important here.

"then i can set like 3 in function and look if its lower then epsilon (0.5) and it is": "like" is not appropriate in mathematical text. Which function: f(x) from (*) above or |f(x) - 2|? How can you check whether |f(x) - 2| < 0.5 for all x > 3, i.e., for infinitely many x? And why would you need to check this if you have just solved this inequality?