# Limit $\lim \limits_{h\to 0} \frac{\sqrt{3x+h}-\sqrt{3x}}{h}$

I'm trying to solve a problem with two variables, but the limit only addresses one. Wolfram Alpha simplifies it for h. Should I leave it simplified, or does x need to be a real number?

$$\lim_{h\to 0} \frac{\sqrt{3x+h}-\sqrt{3x}}{h}$$

• You don't need to use Wolfram Alpha here. Multiply by $$\frac{\sqrt{3x+h}+\sqrt{3x}}{\sqrt{3x+h}+\sqrt{3x}}$$ – oldrinb Sep 8 '13 at 21:08
• What limit exactly are you trying to compute?You didn't mention any variable for the limit. – Git Gud Sep 8 '13 at 21:09
• @oldrinb that's not what I mean. I meant whether or not I could simplify h. – Mia Sep 8 '13 at 21:09
• @GitGud he did, ugly notation though... – AlexR Sep 8 '13 at 21:09
• @MiaCcio what does it mean to 'simplify $h$'? – oldrinb Sep 8 '13 at 21:10

If the limit is as $h$ approaches 0, then the $x$ is not changing, only $h$ is. So, $x$ is treated as if it were a constant in the calculation.
Can you put the limit in the form $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$?
Hint: \begin{align} \lim_{h\to0}\frac{\sqrt{3x+h}-\sqrt{3x}}{h} &=\lim_{h\to0}\frac{\sqrt{3x+h}-\sqrt{3x}}{h}\frac{\sqrt{3x+h}+\sqrt{3x}}{\sqrt{3x+h}+\sqrt{3x}}\\ &=\lim_{h\to0}\frac{(3x+h)-(3x)}{h(\sqrt{3x+h}+\sqrt{3x})} \end{align}