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

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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!
 
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No need to simplify any further, just note what happens as h \to 0. :wink:
 
Supposedly I can simplify it to

1
----
2\sqrt{a}?
 
Which is what you get when you evaluate the limit as h \to 0.
 
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.
 
windwitch said:
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.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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