Limits question

Daniel S

Made a balls of asking how to do it at the end of a thread so here's a clean attempt.

This question is holding me up big time and I really, really need an explaination.

Here's the original question:



(The "=4" should not be there, I fired the whole question into WolframAlpha to try and get step-by-step instructions but as you can see it just gave me the answer.)

From the answer sheet that the lecturer gave me, this is supposed to be the next step before you get the answer:




Lads, I'm really stuck on this and really need help.

Thanks a million!

ray giraffe

Multiply top and bottom by the conjugate!

Michael Collins

With limits involving surds like this you have have to multiply ('above' and 'below') by the conjugate i.e.

[latex] \displaystyle \sqrt{a}-\sqrt{b} \to \left(\frac{\sqrt{a}-\sqrt{b}}{1}\right)\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}, [/latex]

which clearly doesn't change the value of the expression but it will remove the surds from the numerator.

Multiplying out (or by applying the formula for the difference of two squares) you have

[latex] \displaystyle \frac{a-b}{\sqrt{a}+\sqrt{b}}[/latex]

Which is what the second expression is in your post. It should be fairly straightforward then to work out the limit.

Daniel S

Thanks lads!