Question

Finding the limit as x approaches infinity, is there an easier way to find the limit other than graphing it?

Answer 1

Like a sort of equation or something?

Answer 2

It depends on the expression for which you have to find the limit. What is the function?

If, for instance, you have a rational expression with polynomials in both numerator and denominator, such as
\[\frac{P(x)}{Q(x)}=\frac{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}{b_mx^m+b_{m-1}x^{m-1}+\cdots+b_1x+b_0}\]
then you have a few tricks at your disposal.

First, if \(n=m\) (that is, the degrees of the numerator and denominator are the same, then the limit as \(x\to\infty\) will be the ratio of the leading coefficients, \(\dfrac{a_n}{b_m}\).

If \(n>m\), the numerator increases faster than the denominator, so the expression will approach \(\infty\).

If \(m>n\), then denominator will increase faster, and the expression approaches 0.

Answer 3

Keep in mind this is just an example. There's no guarantee this sort of reasoning works for any expression. Give us your specific question and we'll see what we can do.