# Limits

• Dec 28th 2012, 07:35 AM
srirahulan
Limits
$\displaystyle \lim_{x\rightarrow1}\frac{x^2-1}{\sqrt{3x+1}-\sqrt{5x-1}}$ ..How can i Solve this using Theroms??
• Dec 28th 2012, 08:07 AM
zachd77
Re: Limits
I'd suggest multiplying the limit by the conjugate of the bottom (multiply top and bottom by sqrt(3x+1)+sqrt(5x-1)). Work from there and see if you can simplify the limit a bit.
• Dec 28th 2012, 08:20 AM
Soroban
Re: Limits
Hello, srirahulan!

What "theorems" are you referring to?

Quote:

$\displaystyle \lim_{x\to1}\frac{x^2-1}{\sqrt{3x+1}-\sqrt{5x-1}}$

Multiply top and bottom by the conjugate.

. . $\displaystyle \lim_{x\to1}\frac{x^2-1}{\sqrt{3x+1} - \sqrt{5x-1}} \cdot \frac{\sqrt{3x+1} + \sqrt{5x+1}}{\sqrt{3x+1} + \sqrt{5x-1}}$

. . . . $\displaystyle =\;\lim_{x\to1}\frac{(x^2-1)\left(\sqrt{3x+1} + \sqrt{5x-1}\;\!\right)}{(3x+1) - (5x-1)} \;=\;\lim_{x\to1}\frac{(x^2-1)\left(\sqrt{3x+1} + \sqrt{5x-1}\right)}{-2x + 2}$

. . . . $\displaystyle =\;\lim_{x\to1} \frac{(x-1)(x+1)\left(\sqrt{3x+1} + \sqrt{5x-1}\right)}{-2(x-1)} \;=\;\lim_{x\to1}\left[-\frac{(x+1)\left(\sqrt{3x+1} + \sqrt{5x-1}\right)}{2}\right]$

. . . . $\displaystyle =\;-\frac{2(\sqrt{4} + \sqrt{4})}{2} \;=\;-(2+2) \;=\;\boxed{-4}$
• Dec 28th 2012, 03:02 PM
srirahulan
Re: Limits