Find constants a and b that guarantee that the graph of the function defined by f(x)= will have a vertical asymptote at x = 5 and a horizontal asymptote at y = -3.
For a vertical asymptote, when , , so the denominator of
Then we have

The line y = -3 is a horizontal asymptote for f(x), so

So the constants are
(2) Find all vertical tangents and vertical cusps for each of the following functions. Justify your work.
Vertical tangent:
The graph of a function f(x) has a vertical tangent at the point (x0,f(x0)) if and only if

Vertical cusps:
In general we say that the graph of f(x) has a vertical cusp at x0,f(x0)) iff

or

The function is only defined on , so for
First we need to find f'(x)
,
So clearly when , f'(x) does not exist,
When x=-2,
When x=2,
So the two vertical tangent lines are and .

When x<0,
When x>0,
So

Therefore the graph has a vertical cusp at x=0
Here is the graph of f(x).

please see the attached file.