- Thread starter
- #1

- Thread starter bnosam
- Start date

- Thread starter
- #1

- Feb 13, 2012

- 1,704

Why don't multiply numerator and denominator by $\displaystyle \sqrt{x} + \sqrt{a}$?...\(\displaystyle \lim_{x \to a} \frac{ x^2 - a^2}{\sqrt(x) - \sqrt(a)}\)

I've tried to solve this standard, but I either end up with 0 in the denominator, or I end up with 0/0.

Any hints on what to do with this next?

Thanks

Kind regards

$\chi$ $\sigma$

- Thread starter
- #3

I tried that, however I end up with x - a, at the bottom, which leads to 0 in the denominator.Why don't multiply numerator and denominator by $\displaystyle \sqrt{x} + \sqrt{a}$?...

Kind regards

$\chi$ $\sigma$

\(\displaystyle \frac{x^2\sqrt(x) + x^2 \sqrt(a) - a^2\sqrt(x) - a^2\sqrt(a)}{x-a}\)

- Jan 17, 2013

- 1,667

\(\displaystyle (x^2-a^2)=(x-a)(x+a)=(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})(x+a)\)

- Feb 13, 2012

- 1,704

$\displaystyle \frac{x^{2} - a^{2}}{\sqrt{x}-\sqrt{a}}\ \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}} = \frac{x^{2}-a^{2}}{x-a}\ (\sqrt{x}+\sqrt{a})= (x+a)\ (\sqrt{x}+\sqrt{a})$I tried that, however I end up with x - a, at the bottom, which leads to 0 in the denominator.

\(\displaystyle \frac{x^2\sqrt(x) + x^2 \sqrt(a) - a^2\sqrt(x) - a^2\sqrt(a)}{x-a}\)

Kind regards

$\chi$ $\sigma$

- Thread starter
- #6

Ohh ok, I should have seen that.$\displaystyle \frac{x^{2} - a^{2}}{\sqrt{x}-\sqrt{a}}\ \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}} = \frac{x^{2}-a^{2}}{x-a}\ (\sqrt{x}+\sqrt{a})= (x+a)\ (\sqrt{x}+\sqrt{a})$

Kind regards

$\chi$ $\sigma$

\(\displaystyle x\sqrt(x) + x\sqrt(a) + a\sqrt(x) + a\sqrt(a)\)

\(\displaystyle = 4a\sqrt(a)\)

That seem right?

- Moderator
- #7

- Jan 26, 2012

- 995

It should technically be \[\color{red}{\lim\limits_{x\to a}} x\sqrt{x}+x\sqrt{a}+a\sqrt{x}+a\sqrt{a} = 4a\sqrt{a}\]Ohh ok, I should have seen that.

\(\displaystyle x\sqrt(x) + x\sqrt(a) + a\sqrt(x) + a\sqrt(a)\)

\(\displaystyle = 4a\sqrt(a)\)

That seem right?

since you're supposed to include the $\displaystyle\lim_{x\to a}$ part in each line of your work leading up to the substitution of $x=a$ at the end of the problem.

Otherwise, everything looks fine to me!

- Admin
- #8

Just a $\LaTeX$ tip:\(\displaystyle \lim_{x \to a} \frac{ x^2 - a^2}{\sqrt(x) - \sqrt(a)}\)

I've tried to solve this standard, but I either end up with 0 in the denominator, or I end up with 0/0.

Any hints on what to do with this next?

Thanks

Use the code \sqrt{x} instead of \sqrt(x) and the argument will be put under the radical, to get $\sqrt{x}$ instead of $\sqrt(x)$.