Question 1052647
Why?  What happens NEAR the asymptotes?  


The best way to figure how to sketch a graph is to identify the critical x values.  Those would be any zeros of g(x) and any vertical asymptotes.  These cut the x-axis into intervals for you to check how g(x) behaves.

<pre>
<b>x value        significance</b>
-3             vertical asymptote
0              root or zero
3              vertical asymptote
</pre>

Next look at the signs for g in each interval on x.  Pick any x value in each interval.
<pre>
INTERVAL             pick x          sign of g
(-infinity,-3)       -4              -5(-4)/((-4-3)(-4+3))=(+)/((-)(-))=POSITIVE
(-3,0]               -1              -k(-1)/((-)(+))=NEGATIVE
[0,3]                 1              -k(+)/((-)(+))=POSITIVE
[3,infinity)          10             -k(+)/((+)(+))=NEGATIVE
</pre>



Think about g in those four intervals.
Starting very far from the left, approaching -3, g is positive, changing from very small to very large, and large without bound  toward this asymptote.  
Immediately to the right of this x=-3 asymptote, suddenly g is very large BUT NEGATIVE...,   Do you see how this works?  

Also notice change from negative to positive as g crosses the x-axis at 0.
,...



{{{graph(400,400,-12,12,-12,12,-5x/(x^2-9))}}}



The horizontal asymptote is y=0.