What is the limit of the form 0/0?

[Victim]

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.pleasez help me.[/B]

 

[Buzz Bloom]

The attempt at a solution i had done rationalisation but the form is not simplifying.pleasez help me.

Hi Victim:

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

Hope this helps.

Regards,
Buzz

 

[Victim]

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]

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

 

[Victim]

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]

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

 

[Victim]

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]

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.

 

[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 !