# Thread: Help with factoring this limits equation by rationalizing.

1. ## Help with factoring this limits equation by rationalizing.

So yeah trying to factor this and this is probably my algebra being rusty coming back to bite me in the ass.

lim x -> 2 for the function

3 - square root of 5x - 1 / x- 2

so this is as far as I got

(3 - square root of 5x - 1)(3 + square root of 5x-1) / (x - 2) (3 + square root of 5x - 1)

9 - (5x - 1) / (x - 2)(3 + square root of 5x - 1)

and I'm stuck.

Sorry if there is a way to insert a square root symbol I'm brand new to this site.

2. ## Re: Help with factoring this limits equation by rationalizing.

Is it \displaystyle \begin{align*} 3 - \sqrt{5}\,x - \frac{1}{x} - 2 \end{align*} as written? If not, please add some brackets so we know exactly what you mean...

3. ## Re: Help with factoring this limits equation by rationalizing.

Sorry its

{3 - [square root 5x - 1]} / (x - 2)

4. ## Re: Help with factoring this limits equation by rationalizing.

Originally Posted by XxXScorpionXxX
Sorry its

{3 - [square root 5x - 1]} / (x - 2)
so you are trying to find

$\displaystyle{\lim_{x\to 2}}~\dfrac{3-\sqrt{5x-1}}{x-2}$ ?

$\dfrac{3-\sqrt{5x-1}}{x-2}=\dfrac{3-\sqrt{5x-1}}{x-2} \cdot \dfrac{3+\sqrt{5x-1}}{3+\sqrt{5x-1}}=$

$\dfrac{9-(5x-1)}{(x-2)(3+\sqrt{5x-1})} = \dfrac{10-5x}{(x-2)(3+\sqrt{5x-1})} =$

$\dfrac{-5(x-2)}{(x-2)(3+\sqrt{5x-1})} = \dfrac{-5}{(3+\sqrt{5x-1})}$

$\displaystyle{\lim_{x\to 2}}~\dfrac{-5}{(3+\sqrt{5x-1})}=-\dfrac 5 6$

5. ## Re: Help with factoring this limits equation by rationalizing.

$\lim\limits_{x\to2}=\dfrac{3-\sqrt{5x-1}}{x-2}$

When we substitute $x=2$ in the function then we get indeterminate form $(\frac{0}{0})$ so, applying L hospital's rule-

$\lim\limits_{x\to2}=\dfrac{\dfrac{\mathrm d}{\mathrm dx}(3-\sqrt{5x-1})}{\dfrac{\mathrm d}{\mathrm dx}(x-2)}$

$\lim\limits_{x\to2}=\dfrac{\bigg(\frac{-1×5}{2\sqrt{5x-1}}\bigg)}{1}$

$\lim\limits_{x\to2}=\dfrac{-5}{2\sqrt{5x-1}}$

$=\dfrac{-5}{2\sqrt{5×2-1}}$

$=-\dfrac{5}{6}$

6. ## Re: Help with factoring this limits equation by rationalizing.

Yes, L'Hopital's rule works here but on a problem like this I would consider "overkill". Like using a shot gun to swat a fly.

7. ## Re: Help with factoring this limits equation by rationalizing.

thank you, I feel kinda dumb, now it seems obvious.