# How to use linearization at the point where the given function is not defined

Let's say we are given a function $$f(x)$$, which is not defined at the point $$x_0$$. How do we find linear approximation of $$f$$ near $$x_0$$? P.S. I wrote "linear" just to make things simpler, I came across this problem while trying to approximate the following function near zero: $$\frac{lnx}{x*e^x}$$. My problem is that to approximate this function near zero I have to put zero for x in the function as the first (or zeroth) term of Maclaurin series, but this way I get zero in the denominator.

For example if you want to evaluate a function $$f$$ at some point $$x_0$$ where it not defined, you could try evaluating the function in limit as $$x \to x_0$$. Now in your specific case this will not work as function obviously explodes as it approaches zero. Reason for this is as x goes to zero, the logarithm goes to negative infinity and $$\frac{1}{x}$$ goes to zero, thus the fraction goes to negative infinity. Further your function is not defined to the left of zero, and there is no way to build a derivative.
Best thing you can hope for is to linearize your function around $$0 + \varepsilon$$.
Usually even if function is undefined at some point, but is defined on both sides of the point you can use the $$\lim \limits_{h\to 0} \frac{f(x+h) - f(x-h)}{2h}$$ definition of limit to evaluate the derivative at that point.