Thread: Finding the domain and asymototes of a function

Hi..I'm not completely clueless on this problem I would just like to see if my answers are correct. The problem is: let f be the function given by f(x)=x/sqrt((x^2)-4)
A) find the domain of f
B) find the vertical asymototes of f
C) find the horizontal asymototes of f
When I attempted this problem algebraic ally (which is what the directions said) I got the answer of (-infinity,-2),(2,infinity) for the domain, vertical asymototes at x=2 and x=-2 and the horizontal asymptote to be y=o because the degree of the denominator is bigger than the numerator...however thinking twice about those rules I now think that only applies to rationalized functions..which this is not...annd when I graphed it on my calculator I got a corner hyperbola with what appeared to be a vertical asymptote at x=I and horizontal asymototes at y=1 and y= -1....please help!! Thanks!-Kathryn

Your thinking is on the right track but there is just one small mistake:

because the degree of the denominator is bigger than the numerator

The degrees are in fact equal, because of the square root. In the denominator, as x approaches infinity or negative infinity, the term matters less and less and eventually not at all, as it gets swamped by the increasingly large term, which is square rooted. So essentially, as x approaches positive infinity, the function behaves like . At negative infinity, this is similar, except the numerator will be negative while the denominator will always be positive, so it approaches -1. This accounts for the horizontal asymptotes.

And there are actually vertical asymptotes at , you were correct there.

You did fine for the domain and the vertical asymptotes, for the horizontal asymptotes, consider:





What do you find when you evaluate these limits?

Oh ok..it makes sense now to use limits..thanks so much!!