# Limit of the form 0/0

## Homework Statement

lim x~a 〈√(a⁺2x) -√(3x)〉 ÷ 〈√(3a+x) - 2√x〉[/B]

## Homework Equations

rationalisation and factorisation[/B]

## The Attempt at a Solution

i had done rationalisation but the form is not simplifying.plzz help me.[/B]

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Buzz Bloom
Gold Member
The attempt at a solution i had done rationalisation but the form is not simplifying.plzz help me.
Hi Victim:

One technique is to take the ratio of the derivatives of the numerator and denominator.

Hope this helps.

Regards,
Buzz

Hi Victim:

One technique is to take the ratio of the derivatives of the numerator and denominator.

Hope this helps.

Regards,
Buzz
i need to solve this question without using LH rule,so i had to do it by rationalisation and factorisation.

Buzz Bloom
Gold Member
rationalisation and factorisation
Hi Victim:

I confess that I do not know this terminology. Many decades ago when I was learning math as an undergraduate, this terminology was not used. I also do not understand the notation: x~a.

You describe the problem as determining a value when x->0, which produces the form 0/0. However, I suggest you recalculate the limit of the denominator as x->0.

Regards,
Buzz

Hi Victim:

I confess that I do not know this terminology. Many decades ago when I was learning math as an undergraduate, this terminology was not used. I also do not understand the notation: x~a.

You describe the problem as determining a value when x->0, which produces the form 0/0. However, I suggest you recalculate the limit of the denominator as x->0.

Regards,
Buzz
by x~0 i mean x tends to 0.
rationalising the denominator so that only rational terms are left on the denominator

Delta2
Homework Helper
Gold Member
Because the post doesn't show correctly in my browser, are those expressions under square root? Is it
##\lim_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}##

Because the post doesn't show correctly in my browser, are those expressions under square root? Is it
##\lim_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}##
yes exactly

Delta2
Homework Helper
Gold Member
Ok fine, first multiply both the numerator and the denominator by ##\sqrt{3a+x}+2\sqrt{x}## but do the algebra only in the denominator.

After multiply the fraction that you ll get at the end of the first step, both the numerator and the denominator by ##\sqrt{a+2x}+\sqrt{3x}## but do the algebra only in the numerator.

If all goes well there will be revealed a common factor, which will be simplified and you ll be left with the fraction
##\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}## which has an easy to find limit.

ehild and Victim
Ok fine, first multiply both the numerator and the denominator by ##\sqrt{3a+x}+2\sqrt{x}## but do the algebra only in the denominator.

After multiply the fraction that you ll get at the end of the first step, both the numerator and the denominator by ##\sqrt{a+2x}+\sqrt{3x}## but do the algebra only in the numerator.

If all goes well there will be revealed a common factor, which will be simplified and you ll be left with the fraction
##\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}## which has an easy to find limit.
THANK you so much.you really deserve well !!!

Delta2