# Evaluating a limit using only rationalization and algebraic methods

Evaluate (without using de L'Hôpital Rule)

$$\lim_{x \to -3}\frac{\sqrt{3 x^2+10 x+4}-\sqrt{x^2+3 x+1}}{\sqrt{5 x^2+11 x+5}-\sqrt{2 x^2+x+2}} = \frac 58 \sqrt{17}$$

I have to evaulate the limit of this function as it approaches $-3$. I have tried plugging it in but I get $0/0$. Then I tried LCD, but the $\sqrt x-\sqrt y$ does not equal $\sqrt{x-y}$. Then I tried multiplying the conjugate of the denominator but after all of that I still get $0/0$. I cannot use the calculator except to check my answers after completion. However when I checked there is a limit at $-3$ so I resorted to looking online and it gave me the answer $\frac58\sqrt{17}$, but I cannot seem to get this answer without using L'Hôpital's rule.

• I edited out two inappropriate tags. By the way, what is your question exactly? How to get to that result? Show your effort first! – Ant Sep 26 '14 at 13:10
• Hi and welcome to the site! Since this is a site that encourages learning, you will get much more help if you show us what you have already done. Could you edit your question with your thoughts and ideas? – 5xum Sep 26 '14 at 13:12
• Sorry I am new, but thank you. I have to evaulate the limit of this function as it approaches negative 3. I have tried plugging it in but I get 0/0. Then I tried LCD, but the sqrt(x)-sqrt(y) does not equal Sqrt(x-y). Then I tried multiplying the conjugate of the denominator but after all of that I still get 0/0. I cannot use the Calculator except to check my answers after completion. However when I checked there is a limit at -3 so I resorted to looking online and it gave me the answer (5/8)sqrt(17), but I cannot seem to get this answer w/out using L'Hopital's Rule – Amy Palacios Sep 26 '14 at 13:16
• Have you tried the conjugate of both numerator and denominator? If your original expression becomes $\frac 00$ when $x=-3$ you should be looking to cancel a factor of $x+3=x-(-3)$ from numerator and denominator – Mark Bennet Sep 26 '14 at 13:19

## 1 Answer

Hint 1) $\frac{\sqrt{a}- \sqrt{b}}{\sqrt{c}-\sqrt{d}} = \frac{(a-b)(\sqrt{c} + \sqrt{d})}{(c+d)(\sqrt{a}+\sqrt{b})}$

Hint 2) $2x^2 + 7x +3 = (2x+1)(x+3)$, this is $a+b$

Hint 3) Try hint 2) with $c+d$.

What do you get?

• For c+d I got (3x+1)(x+3) Does that mean I can cancel them out? – Amy Palacios Sep 26 '14 at 13:43
• are you familiar with cancellation rules? – Alex Sep 26 '14 at 13:49
• $\frac{a \cdot b \cdot c}{d \cdot b \cdot e} = \frac{a \cdot c}{d \cdot e}$ – Alex Sep 26 '14 at 13:52
• Thank you, I canceled out the (x+3) and got the answer. However, how did you know that rule you gave in hint 1 – Amy Palacios Sep 26 '14 at 14:00
• It's just a lot of practice, Amy... – Alex Sep 26 '14 at 14:05