Limit help!

[HelpmeHelpyou6]

Lim sqrt(2(a+h)) - sqrt(2a)/h
h-->0
Find in terms of constant a:


No matter what I do, i just keep getting zero! Help!

 

[pka]

Find in terms of constant a: Lim sqrt(2(a+h)) - sqrt(2a)/h
h-->0

\(\displaystyle \displaystyle\sqrt{2(a+h)}-\sqrt{2(a)}\frac{\sqrt{2(a+h)}+\sqrt{2(a)}}{\sqrt{2(a+h)}+\sqrt{2(a)}}\)\(\displaystyle =\dfrac{2h}{\sqrt{2(a+h)}+\sqrt{2(a)}}\)

 

[pappus]

Lim sqrt(2(a+h)) - sqrt(2a)/h
h-->0
Find in terms of constant a:


No matter what I do, i just keep getting zero! Help!

1. I assume that you mean

\(\displaystyle \lim_{h \to 0} \left(\frac{\sqrt{2(a+h)}-\sqrt{2a}}{h} \right)\)

If sO:

2.

\(\displaystyle \lim_{h \to 0} \left(\frac{\sqrt{2(a+h)}-\sqrt{2a}}{h} \right) = \lim_{h \to 0} \left(\frac{\sqrt{2(a+h)}-\sqrt{2a}}{h}\cdot \frac{\sqrt{2(a+h)}+\sqrt{2a}}{\sqrt{2(a+h)}+\sqrt{2a}} \right)\)

\(\displaystyle \lim_{h \to 0} \left(\frac{2h}{h \cdot \left( \sqrt{2(a+h)}+\sqrt{2a} \right)} \right)\)

Cancel h.

3. Now your limit looks like:

\(\displaystyle \lim_{h \to 0} \left(\frac{2}{ \sqrt{2(a+h)}+\sqrt{2a}} \right)\)

If h approaches zero \(\displaystyle \sqrt{2(a+h)}\) approaches \(\displaystyle \sqrt{2a}\)

So you'll get:

\(\displaystyle \lim_{h \to 0} \left(\frac{2}{ \sqrt{2(a+h)}+\sqrt{2a}} \right) = \frac{2}{ \sqrt{2a}+\sqrt{2a}} = \frac{2}{2\sqrt{2a}} = \frac1{\sqrt{2a}}\)