How do you find the end behavior of # [(x–1)(x+2)(x+5)] / [x(x+2)2]#?

1 Answer
Nov 18, 2015

Answer:

#\lim_{x\to\pm\infty} \frac{(x-1)(x+2)(x+5)}{2x(x+2)} = \pm\infty#

Explanation:

First of all, the expression can be simplified, since a factor #(x+2)# appears in both numerator and denominator. So, we have

#\frac{(x-1)cancel((x+2))(x+5)}{2xcancel((x+2))} = \frac{(x-1)(x+5)}{2x}#

From here, we can expand the numerator:

#\frac{(x-1)(x+5)}{2x} = \frac{x^2+4x-5}{2x}#

To study the end behavior, simply factor our the highest power in both numerator and denominator:

# \frac{x^2+4x-5}{2x} = \frac{x^2(1+4/x-5/x^2)}{2x} = \frac{x(1+4/x-5/x^2)}{2}#

Now, as #x\to\pm\infty#, you have that #4/x# and #-5/x^2# tend to zero. So, the whole expression has the same behaviour as #x/2# (which again is the same behaviour as #x#).

So, since #\lim_{x\to\pm\infty} x = \pm\infty#, this will also be the behaviour of your expression.