x/[(1+3x)^(1/2)-1] I'm wondering if it's even possible to imagine what a graph of this fxn would look like, or do you definitely need a graphing calc? When I plug it into a TI83, it ends up looking pretty linear and apparently doesn't exist when x < 0 ~Jules~ PS. sorry for how messy the eq'n looks, it's difficult to type. ... (1+3x) should be under a sqrt. sign.
Just by looking at it, you see that you have x in the numerator and x^{1/2} in the denominator. For large x, the function will roughly look like x^{1/2}, but you can find a better approximation. After multiplying the top and bottom of the fraction by the conjugate of the denominator and doing some work on it, you get [tex]\frac{\sqrt{3x + 1} + 1}{3} = \frac{\sqrt{3x(1 + \frac{1}{3x})} + 1}{3} = \frac{\sqrt{3x}\sqrt{1 + \frac{1}{3x}}}{3} ~+~ \frac{1}{3} = \frac{\sqrt{3}}{3}\sqrt{x} \sqrt{1 + \frac{1}{3x}} ~+~ \frac{1}{3}[/tex] As x→∞, the larger radicand goes to 1 and has less and less effect on √x, so the function is nearly like [tex]\frac{\sqrt{3}}{3}\sqrt{x} ~+~ \frac{1}{3}[/tex]