Question

How do you evaluate the limit `(x^2+4x+6)/(3x^2+1)` as x approaches `oo`?

Answer 1

`1/3`. Just ignore all the terms except the highest power top and bottom.

Explanation:

If the function is the quotient of two polynomials `{P(x)}/{Q(x)}` where the degree of `P` is the same as that of `Q` the limit will always be the ratio of the coefficients of the highest power. If the degree of `Q` is higher than that of `P` the limit will be zero; if the degree of `Q` is lower that that of `P` there will be no limit. Informally, as `x` gets larger and larger all the non-first terms of `P` and `Q` become negligible compared to the first.