Hello there I have a question that I am trying to answer for

Hello there, I have a question that I am trying to answer for Precalculus:

I came up with the graphs looking like this:

Is the similarity that their slopes are the same? Any assistance you can offer is much appreciated! Thanks!!

Solution

There are similarities in the two functions for very x >>0 and x<< 0

We can see g(t) = 3t ( straight line with slope =3)

h(t) = 3t + 2/t has different nature as compared to g(x) before y =5 and y =-5

because of the term 2/t

however for t <<0 eg: t = - 1000 ; h(t) = 3(-1000) + 2/1000 = -30000 - 0.002

So, the term 3t is more dominant.

for t >>0 t = 1000 ; h(t) = 30000 +0.002

So, the term 3t is more dominant

We can say the end behaviuor of h(t) is simlar to h(t) = 3t which same as g(t)