Give an equation with a vertical asymptote of 0 and a slant asymptote of -x
The equation of the slant asymptote is put in the standard form and it is written as:
y = mx + n, where m is the slope of the asymptote and n is the y intercept.
The equation of the slant asymptote is:
y = -x
m = -1 and n = 0
The equation y = -x is the equation of the bisectrix of the 2nd or the 4th quadrant.
So, the slant asymptote y = -x is the bisectrix of the 2nd or the 4th quadrant.
If the vertical asymptote of a function is x = 0 that means that the function is discontinuous in the point that has the x coordinate x = 0.
The domain of definition of the function that has the vertical asymptote x = 0 does not contain the value x = 0.
To give an example of a function whose vertical asymptote is 1/x and the obleque asymptote is y = x.
We know that an asymptote is a line or a curve to which a function approach as x --> infinity (or in limit).
We know that y = 1/x - x is an example.
We discuss below how this is possible.
As x--> infinity y = 1/x-x approaches y = -x , as 1/x approaches zero.
As x --> 0, y = 1/x -x approaches y = 1/x which becomes unbounded.
So x= 0 is the vertical asymptote and y = -x is the oblique asymptote.