How Do You Further Simplify the Limit of a Square Root Expression?

[pyrosilver]

Homework Statement

lim
_{}h\rightarrow0 (\sqrt{a+h} - \sqrt{a}) / h

3. ok so i multiplied by \sqrt{a+h} + \sqrt{a}. The top simplified to a+ h -a, aka h. the bottom is h(\sqrt{a+h} + \sqrt{a}).

The h cancels, so I am left with

1 / (\sqrt{a+h} + \sqrt{a}

how do i simplify this further?

thanks!

 

[jgens]

No need to simplify any further, just note what happens as h \to 0.

 

[pyrosilver]

Supposedly I can simplify it to

1
----
2\sqrt{a}?

 

[jgens]

Which is what you get when you evaluate the limit as h \to 0.

 

[windwitch]

It makes perfect sense as well that that is your answer. I am not sure of the extent of your understanding of differentiation, however, if you know enough, you should be able to recall that the derivative of a^(1/2) = 1/2a^(1/2) which is exactly what you got.

 

[Mark44]

It makes perfect sense as well that that is your answer. I am not sure of the extent of your understanding of differentiation, however, if you know enough, you should be able to recall that the derivative of a^(1/2) = 1/2a^(1/2) which is exactly what you got.

A clearer way of writing this would be a^(1/2) = 1/(2a^(1/2)). A literal reading of what you wrote would be 1/2*a^(1/2); that is, the fractional power would be in the numerator, which I'm sure you didn't intend.