given the function:
f(x) = (x2) / (x-1)
the correct answer to the limit of f(x) as x approaches infinity is:
y = x+1
all math references point to this answer and the method they all use is long division of x-1 into x2
however if one were to multiply both the numerator and denominator by 1/x and then take the limit, one gets:
how can the descrepency between the two answers be explained?
Your approach of multiplying numerator and denominator by 1/x shows that the limit of x2/(x-1) goes to infinity as x goes to infinity. There's really not much difference between infinity and (infinity + 1). To show that y = x+1 is an asymptote of the curve c: y = x2/(x-1) as x goes to infinity you need to show that the graph of the line y = x+1 approaches the graph of the curve, which it does. If you don't believe me, look at the difference (for fixed x) between x-1 and c, and you will see that the difference approaches 0 as x goes to infinity. This is the argument given in the math references you saw. The difference between x and c approaches 1.
Other methods: You can
(a) let y = x-1 (so that x = y+1) and change the given function into an expression involving y that is easier to divide by. (You'll find that the limit is y+2, which equals x-1.)
or (b) Put the expression ax + b - [x2/(x-1)] over a common denominator, then set the numerator = 0 and solve for a and for b. (You'll find that a=1 and b=1.)Chris and Penny