# the limit as x approaches infinity

• Sep 14th 2008, 05:51 PM
thecount
the limit as x approaches infinity
I know the answer to this problem, but I do not know how to get to it.

lim (sqrt(9x^2+x)-3x) = (1/6)
x-> infinity
• Sep 14th 2008, 05:57 PM
o_O
Multiply top and bottom by its conjugate: $\displaystyle \sqrt{9x^2 + x} {\color{red}+} 3x$:

$\displaystyle \lim_{x \to \infty} \left(\sqrt{9x^2 + x} - 3x\right) \cdot \frac{{\color{red}\sqrt{9x^2 + x} + 3x} }{{\color{red}\sqrt{9x^2 + x} + 3x}}$

The numerator is an application of the difference of squares formula: $\displaystyle (a-b)(a+b) = a^2 - b^2$

So: $\displaystyle = \lim_{x \to \infty} \frac{9x^2 + x \ - \ 9x^2}{\sqrt{9x^2 + x} + 3x}$

etc. etc.
• Sep 14th 2008, 06:01 PM
skeeter
$\displaystyle \frac{\sqrt{9x^2 + x} - 3x}{1} \cdot \frac{\sqrt{9x^2 + x} + 3x}{\sqrt{9x^2 + x} + 3x}$

$\displaystyle \frac{(9x^2 + x) - 9x^2}{\sqrt{9x^2 + x} + 3x}$

$\displaystyle \frac{x}{\sqrt{9x^2 + x} + 3x}$

divide every term by x ...

$\displaystyle \frac{1}{\sqrt{9 + \frac{1}{x}} + 3}$

now let $\displaystyle x \to \infty$
• Sep 15th 2008, 01:06 PM
Moo
Hello,

Another approach is to use asymptotic equivalences (but I don't know your level, so maybe you can't use it...)

$\displaystyle \sqrt{9x^2+x}-3x=3x \left(\sqrt{1+\frac{1}{9x}}-1\right)$

Since $\displaystyle x \to \infty$, we have $\displaystyle \frac{1}{9x} \to 0$

$\displaystyle \sqrt{1+\frac{1}{9x}}=\left(1+\frac{1}{9x}\right)^ {1/2} \sim_{\substack{x \uparrow \infty}} 1+\frac 12 \times \frac{1}{9x}+ \mathcal{O}\left(\frac{1}{x^2}\right)$

So $\displaystyle \lim_{x \to \infty} \sqrt{9x^2+x}-3x=\lim_{x \to \infty} 3x\left(1+\frac 12 \times \frac{1}{9x}+ \mathcal{O}\left(\frac{1}{x^2}\right)-1\right)$

$\displaystyle =\lim_{x \to \infty} 3x \left(\frac 12 \times \frac{1}{9x}+\mathcal{O} \left(\frac{1}{x^2}\right)\right)=\lim_{x \to \infty} \frac 16+\mathcal{O} \left(\frac 1x\right)=\boxed{\frac 16}$