# Sketching x/[(1+3x)^2-1]

by Jules18
Tags: 3x21, sketching, x or 1
 P: 867 Just by looking at it, you see that you have x in the numerator and x1/2 in the denominator. For large x, the function will roughly look like x1/2, 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 $$\frac{\sqrt{3x + 1} + 1}{3} = \frac{\sqrt{3x(1 + \frac{1}{3x})} + 1}{3} = \frac{\sqrt{3x}\sqrt{1 + \frac{1}{3x}}}{3} ~+~ \frac{1}{3} = \frac{\sqrt{3}}{3}\sqrt{x} \sqrt{1 + \frac{1}{3x}} ~+~ \frac{1}{3}$$ As x→∞, the larger radicand goes to 1 and has less and less effect on √x, so the function is nearly like $$\frac{\sqrt{3}}{3}\sqrt{x} ~+~ \frac{1}{3}$$