Expert: Socrates - 1/1/2005

Question
Let f be the function given by f(x)=x over the sqare root of x^2-4.

a. find the domain of f.

b. write an equation for each vertical asymptote to the graph of f.

c. write an equation for each horizontal asymptote to the graph of f.

d. find the derivative of f(x).

Answer
a) The domain is {x| x>2 or x<-2}, so x can be any number greater than 2 or less than -2. This is because we can't have zero in the denominator or the square root of a negative number.

b) Vertical asymptotes occur where the denominator is zero.
For this function, that happens when x=2 or x=-2, so those are the equations of the vertical asymptotes.

c) Horizontal asymptotes are found by taking the limit as x goes to plus or minus infinity. The limit at plus infinity is 1 , so y=1 is the horizontal asymptote as x goes to plus infinity. The limit as x goes to minus infinity is -1 , so y=-1 is the horizontal asymptote as x goes to minus infinity

d) f'(x) =

((x^2 - 4)^1/2  - (x)(1/2)(2x)(x^2 - 4)^-1/2) / (x^2 - 4)

= -4/(x^2 - 4)^3/2