The first thing one would try
is to just put x= 9/8. Of course, that doesn't work here (it never does for interesting problems) but it shows where the problem is- the denominator becomes 0 while the numerator does not. Now, what happens if x is very close
to 9/8 but just slightly less? You can answer that by actually putting in such a number. 9/8= 1.125 so what happens if x= 1.120? In that case, the numerator is -22x= -24.26 and the denominator is 9- 8(1.120)= 9- 8.96= .04. -24.26/.04= -606.5. In other words, the fraction itself is a very large negative
number- since the denominator is going to 0 and the numerator isn't, the limit will be "negative infinity". You could also do that by recognizing that
. If x is less
than 9/8, the quantity in the parentheses,
is negative so
is the product of two negative numbers and so is negative. Since the numerator, -22x, is also negative, the fraction is negative and the limit is negative infinity. For x greater
so that the denominator is negative and the fraction is the quotient of two
negative numbers. The fraction is positive so the limit is positive
multiply by the fraction
then do thesame thing.
The horizontal asymptotes of the curve are, of course, the values of y as x goes to plus or minus infinity. I recommend that you divide both numerator and denominator by x. That way the numerator is the constant 18. What happens in the denominator?