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

Question

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.

Answer 1

Well there are two problems here. Assuming you want to linearize this function in the usual sense of approximating it by first derivative, then you can use a kind of limiting process. In order to evaluate function where it is not formally defined you can use the limiting value.

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.

In your case you could say that at zero, the best "linear" approximation to your function is a vertical line through origin.