evaluating limits

[wendywoo]

Evaluate lim (x/??? x^2-2) as x goes into infinity.

I don't remember what to do if the bottom is a radical.

 

[galactus]

Since $\displaystyle x\to \infty$, get rid of the 2 and write as:

$\displaystyle \lim_{x\to \infty}\frac{x}{\sqrt{x^{2}}}$

$\displaystyle \lim_{x\to \infty}\frac{x}{|x|}$

Since we are heading in the positive direction toward infinity:

$\displaystyle \lim_{x\to \infty}\frac{x}{x}=1$

That's all it is.

You can do this when $\displaystyle x\to \pm\infty$

 

[soroban]

Hello, wendywoo!

Another approach . . .

$\displaystyle \text{Evaluate: }\;\lim_{x\to\infty}\frac{x}{\sqrt{ x^2-2}}$

Divide numerator and denominator by $\displaystyle x$

. . $\displaystyle \lim_{x\to\infty}\frac{\frac{x}{x}}{\frac{\sqrt{x^2-2}}{x}} \;=\;\lim_{x\to\infty}\frac{1}{\sqrt{\frac{x^2-2}{x^2}}} \;=\;\lim_{x\to\infty}\frac{1}{\sqrt{1-\frac{2}{x^2}}} \;=\;\frac{1}{\sqrt{1-0}} \;=\;1$