How to remove the periphery of singularities when plotting a graph of a function

Consider drawing a graph of a function that is positively represented by the form y=f(x) on the (x,y) plane.However, let's say we plot only in the range x_min xx xx_max.

To be simple, you can change x little by little from x_min to x_max and calculate (x, f(x) and connect these points with line segments to draw a graph.

However, with this method, it doesn't work well for graphs with singularities.For example, the inverse graph y=1/x is not defined at x=0.If you use Scratch to implement and run the simple algorithm above, you'll see a vertical line near the y-axis that doesn't exist (the bold green line is the graph).

This is the result of an attempt to draw a graph of

(The following is a script for drawing this graph.Click to enlarge.)

Source code for Scratch to draw a graph of

If you let WolframAlpha draw a graph, it is well plotted except near singularities

In fact, looking at Wolfram Mathematica features, it seems to have detected and removed singularities.

Also, if y=1/x, x=0 is a singularity by manual calculation, so you just need to remove it manually, but it would be troublesome if more than one singularity such as y=tan(x) appears.

Given that the definition expression of function f is given as input, is there an algorithm that prevents you from plotting around singularities well when plotting a graph of y=f(x)? It can be a numerical analysis or a definition expression.If necessary, you can limit the shape of f.Personally, I would be satisfied if I could draw about y=1/x or y=tan(x).I want to make it versatile to some extent, so I would like to know other ways besides manually calculating singularities.

*It was easy to write, so I wrote it in Scratch above, but I just want to know the algorithm, so I don't mind if it's a programming language, a pseudo code, or a policy.