Expert: Josh - 6/2/2005

Question
The graph of y=x/(1-absolute value(x)) has
A. no horizontal asymptotes and one vertical asymptote
B. one horizontal asymptote and one vertical asymptote
C. two horizontal asymptotes and one vertical asymptote
D. one horizontal asymptote and two vertical asymptotes
E. two horizontal asymptotes and two vertical asymptotes
From my graphing calculator I can see the answer is E,but how would you know that without a graphing calculator? How do you do it algebraically?

Answer
Hi Jeff,

Haven't heard from you for a while.

It is usually helpful considering all possible x values which take the function to zero or infinity (in the limit as x approachs some value, say, x[i]).

In particular, if the denominator converges to zero, we always get a vertical asymptote.

Case 1a:
Let x approaches x=-1 from x=0,y=0 (the origin).
The denominator 1-|x| approaches 0 while remaining positive. However, the numerator "x" is negative. Dividing a negative number by a tiny positive quantity takes us towards negative infinity.
So, y->-infinity, as x->-1 from the right hand side (RHS), i.e., x=-1+delta.

Case 1b:
Let x approaches x=+1 from x=0,y=0 (the origin).
The denominator 1-|x| approaches 0 while remaining positive. This time, the numerator "x" is also positive. So, y->+infinity, as x->+1 from the left hand side (LHS), i.e., x=1-delta.

Thus far, we have identified two vertical asymptotes.

Next, consider the dominant behavior, i.e., the polynomial expressions in terms of powers of x. Observe that whenever x goes to positive infinity or negative infinity, the number "1" becomes negligible. So, the x terms dominate the expression and y steadily approaches a limiting value.

Case 2a:
As x goes towards +infinity from x=+1+delta, lim x/(1-|x|) converges to lim |x|/(-|x|). The limit is given by y=-1.

Case 2b:
As x goes towards -infinity from x=-1-delta, lim x/(1-|x|) converges to lim -|x|/(-|x|). The limit is given y=+1.

This completes the analysis.