The vertical asymptotes of a curve are lines that the graph of the curve approaches but does not touch.
For `y = (f(x))/(g(x))` , the vertical asymptotes are lines x = a where a is the root of the denominator g(x).
The equation of the curve in the problem is `y = (1 + x^2)/(3x - 2x^2)`
To determine the roots of the denominator solve 3x - 2x^2 = 0
x(3 - 2x) = 0
x = 0, x = 3/2
The roots of the denominator are x = 0 and x = 3/2.
The vertical asymptotes of the curve `(1 + x^2)/(3x - 2x^2) ` are x = 0 and x = 3/2.
This can be verified by looking at the graph of the curve:
For a curve y = f(x), there are many values of x at which y does not have a real value. If a graph of the curve y = f(x) is drawn, at particular values of x there is a hole. If a vertical line can be drawn here it is called vertical asymptote.
Consider the graph of the curve that is given y = (1+x^2)/(3x - 2x^2)
Let us zoom in at where x = 0,
It can be seen that there is no point on the graph at which x = 0
Similarly let us zoom in at x = 1.5
Again notice that the graph does not have a point where x = 1.5
This shows that the vertical asymptotes of the graph are at x = 0 and x = 1.5
A vertical asymptote is an invisible line (usually depicted as dashed lines) that approach the function without crossing into it.
To calculate the vertical asymptote of the given function, (1+x^2)/(3x-2x^2), simply find the zeros of the denominator. In this case, the zeros of the denominator are:
x=0 and x = 3/2 (or 1.5)