I am not sure exactly what information you want here.
The asymptotes as x ->∞ will depend on f(x).
If f(x) is bounded, then the asymptote will be y=0.
If f(x) is unbounded, then the limit will look more like the numerator, e.g.
if f(x)=x2 + 2x +1, the asymptote will look like y=x.
As far as limits as x -> -∞, they work the same way, just at the left end of the graph instead of the right.
Paul M.
tutor
First, yes there can only be one asymptote as x approaches + infinity. As you correctly noted it may be a horizontal asymptote (most often y=0) or an asymptote with a slope (e.g f(x)=[(x^2)+1]/x]. I have never seen the methodology you use, but it is certainly correct. However, I would urge you to look at graphs to be sure you can connect the functions with the graphs.
Of course, there can also be an asymptote as x approaches - infinity...but only one in that direction also.
Your computations are correct.
Lastly, remember that there are also vertical asymptotes: look at the graph of y = tan x or any rational function where the denominator is 0.
I hope this helps.
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05/30/20
Io L.
To find oblique asymptote, we can determine k by k = lim x→+∞ [ f(x) / x ] . After that, we can find b= lim x→+∞ [ f(x) -kx] and finally, fulfill y=kx+b that is formula of oblique or horizontal asymptote. Mentioned formula is giving only 1 value for k, so it will be only horizontal or oblique asymptote. My question is - can one function has both oblique and horizontal asymptote and why are we considering only +∞ in mentioned formula of k. For instance, f(x)=(4x^2- 1)^1/2 - x. For this function, K=1 (if lim -> +∞) and k=-3 (if lim x->-∞). So, there are two value for k, does it means that function has two oblique asymptotes?05/30/20