# Thread: limits of a fraction

1. ## limits of a fraction

how do you find the limits of [(1+x) ^(1/2) -1 ] / [ (1-x)^(1/2) -1 ]?

2. Originally Posted by alexandrabel90
how do you find the limits of [(1+x) ^(1/2) -1 ] / [ (1-x)^(1/2) -1 ]?
x tends to?

3. oh sorry! i missed that out..it is as x tends to 0.

4. 0/0 so use L'Hopital's Rule

5. $\displaystyle \displaystyle = \lim_{x \to 0} \frac{\Big(1+\frac{x}{2}-\frac{x^{2}}{8} + O(x^3)\Big) - 1}{\Big(1- \frac{x}{2} - \frac{x^{2}}{8} + O(x^{3})\Big)- 1}$

$\displaystyle \displaystyle = \lim_{x \to 0} \frac{\frac{1}{2} - \frac{x}{8} + O(x^{2})}{-\frac{1}{2} - \frac{x}{8} + O(x^{2})} = -1$

6. Originally Posted by alexandrabel90
how do you find the limits of [(1+x) ^(1/2) -1 ] / [ (1-x)^(1/2) -1 ]?
If you multiply numerator and denominator by (1+x)^(1/2)+ 1 and multiply numerator and denominator by (1- x)^(1/2)+ 1 you get
$\displaystyle \left(\frac{(1-x)^{1/2}+1}{(1+x)^{1/2}+ 1}\right)\left(\frac{1+ x- 1}{1-x- 1}\right)$
$\displaystyle = -\frac{(1-x)^{1/2}+ 1}{(1+x)^{1/2}+ 1}$
and now you can take x= 0.