What is the limit as #x -> ∞# of #(x^2 + 2) / (x^2 - 1)#?

2 Answers
Topscooter
Dec 17, 2015

The answer is #1#.

Explanation:

There is a useful property of rational functions : when #x rarr prop# the only terms that will matter are the terms at the highest degree (which makes perfect sense when you think about it).

So as you can guess, #2# and #-1# are nothing compared to#prop# so your rational function will be equivalent to #x^2/x^2# which is equal to #1#.

George C.
Dec 17, 2015

#lim_(x->oo) (x^2+2)/(x^2-1) = 1#

Explanation:

Here are a couple more ways to look at this:

#lim_(x->oo) (x^2+2)/(x^2-1)#

#= lim_(x->oo) ((x^2-1)+3)/(x^2-1)#

#= lim_(x->oo) (1+3/(x^2-1))#

#= 1 + 0 = 1#

since #3/(x^2-1) -> 0# as #x->oo#

Alternatively, divide both numerator and denominator by #x^2# as follows:

#lim_(x->oo) (x^2+2)/(x^2-1)#

#=lim_(x->oo) (1+2/x^2)/(1-1/x^2)#

#=(1+0)/(1-0)#

#=1#

since #2/x^2 -> 0# and #1/x^2 -> 0# as #x->oo#

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