Realizing oblique asymptotes for non linear functions

Question

I have the function $f(x) = \frac{x^2-x+e}{(lnx)^2}$. The graph seems to have an oblique asymptote. Is this an oblique asymptote or is it just the nature of the function? If yes, how do I find oblique asymptotes for nonlinear functions?

Answer 1

This graph does not have an oblique asymptote. When you take $\lim_{x\rightarrow \infty}\frac{(x^2-x-e)}{(lnx)^2}$= $\lim_{t\rightarrow \infty}\frac{(e^{2t}-e^t-e)}{(t)^2}$, you have an e-power (squared) in the numerator against a quadratic polynomial in the denominator. That quotient can not yield anything linear (with a remainder that goes to zero for large $t$). The graph is misleading, as is often the case when determining asymptotic behavior.