Evaluating the limit using the definton of a limit...
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pka
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First you missed a minus sign. It should be
\(\displaystyle \dfrac{\frac{-h}{x(x+h)}}{h}=\dfrac{-1}{x(x+h)}\) divide all by \(\displaystyle h\).
HallsofIvy
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To divide fractions, invert the denominator and multiply. (I bet you learned that long ago!)
Here the denominator is "h". Inverting it gives \(\displaystyle \frac{1}{h}\).
\(\displaystyle \frac{\frac{h}{x(x+h)}}{h}= \frac{h}{x(x+h)}\frac{1}{h}\)
Cancel.
Here the denominator is "h". Inverting it gives \(\displaystyle \frac{1}{h}\).
\(\displaystyle \frac{\frac{h}{x(x+h)}}{h}= \frac{h}{x(x+h)}\frac{1}{h}\)
Cancel.
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I'm not sure why it would be -h. Once I change the top part of the fraction to a common denominator, isn't the top part then... x - (x + h)? Which I thought that the x's would cancel each other out?
The rest I under stand when you said to divide by h, thank you for that. I'm just confused about that -h?
I'm not sure why it would be -h. Once I change the top part of the fraction to a common denominator, isn't the top part then... x - (x + h)? Which I thought that the x's would cancel each other out?
The rest I under stand when you said to divide by h, thank you for that. I'm just confused about that -h?
I just realized, that when I remove the brackets it then changes the sign to a -h, is that correct?
That expression can be simplified, AND you have not yet taken the limit as h-->0. In fact, if you try to take the limit as written, it won't work because you will get 0/0.