Question

Let f(x)= -3x^4+79x^2-3x+1/ 5x^4+19x^3+2x+5. Discuss the short run behavior for f(x) and the long run behavior for f(x).

Answer 1

The way you type these questions without brackets, it is difficult to determine what the denominator is, as other tutors have also noted.

I will assume that 5x^4+19x^3+2x+5 is your denominator.

To check for vertical asymptotes, I set this equal to zero.
First of all I could not factor it, so I went to a reliable "equation solver".

http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=basic

gave me two complex and two real solutions.
so there are two veritical asymptotes, one around x = -3.8, another around x = -.62
when x=0 your function has value 1/5, so the y-intercept is 1/5

setting the numerator equal to zero in the same program gave me 2 real and 2 complex solutions
the reals at x=-5.15 and x=5.11
so the graph crosses the x-axis at those values
as x approaches ±infinity, your function approaches y=-3/5

(Are you supposed solve these with the use of a programmable calculator?
They seem rather unreasonable to work with otherwise)