End Behavior

1.  Find the end behavior of the following functions:

(a)  Consider:  y = (x²-2x)/(-x³-5x²+4)
as x goes to infinity, y goes to ?
as x goes to – infinity, y goes ?

(b) Consider y = (2x+1)/(-4x+1)
as x goes to infinity, y goes ?
as x goes to – infinity, y goes ?

A:  These are problems that I think it is most important to “think about” as opposed to solve… Look at the first equation:

(a)

y = (x²-2x)/(-x³-5x²+4)

Now think about that whole fraction and what happens as x gets really big… now bigger, now even bigger!  The top is growing at an x² rate and the bottom is growing at an x³ rate.  Most importantly we notice that the bottom is growing at a faster rate than the top.  What happens if you have a fraction where the bottom keeps getting bigger and bigger, but the top doesn’t grow as fast???  Consider 1/100 then 2/1000 then 3/10000 then 4/100000… The fraction is shrinking and shrinking toward zero.

So, in the above equation, as x goes to infinity, y goes to 0.

Now, imagine as x goes to -inifinity.  We still have a top that is growing at an x² and a bottom that is growing at an x³ rate.  Even though we are going to negative infinity, we still have a concept like 1/-100 vs. 2/-1000 vs. 3/-10000….. See how the negative doesn’t really matter and the fraction is still getting smaller and going toward zero?

So, as x goes to -infinity, y goes to 0.

(b)  Now let’s look at the second equation:

y = (2x+1)/(-4x+1)

As x goes to infinity, the +1 on the top and the +1 on the bottom become negligent.  Right?  What is +1 in the scheme of going to infinity?  Useless… So, as x goes to infinity, we are really looking at 2x/-4x

Notice that the exponent on the top x is the same as the exponent on the bottom x (they both have an exponent of 1)… So, the top is essentially growing at the same rate as the bottom….  So, as the x’s get really big, we can cancel them out to get 2/-4 = -1/2

So, as x goes to infinity, y goes to -1/2

The same logic holds for as x goes to -infinity…. y goes to -1/2 as well

I did not show the “math” of this argument, I first just wanted to show the logic.  Make sense so far?

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