# Getting a variable out of the denominator

Discussion in 'Differentiation and Integration' started by mmesford, Dec 30, 2021.

1. ### mmesford

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I’m trying to assess the limit of this term:

[sqrt(x+h) - sqrt(x)]/h

Unfortunately, my basic math skills are failing me. The answer in the book says that the limit of this term as h approaches 0 is 1/2 sqrt(x) but I can’t get there. I’ve tried breaking the fraction into two parts, squaring the top and bottom of each and recombining. This gets me:

h/h^2

That doesn’t help much.

mmesford, Dec 30, 2021

2. ### Country Boy

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You have (sqrt{x+h}- sqrt{x})/h and want to take the limit as h goes to 0.

"Rationalize the numerator" by multiplying both numerator and denominator by sqrt{x+ h}+ sqrt{x}:

(sqrt{x+h}- sqrt{x})(sqrt{x+h}+sqrt{x})/h(sqrt{x+h}+ sqrt{x})= (x+h- x)/(sqrt{x+h}+ sqrt{x})h)= h/(sqrt{x+h}+ sqrt{x})h)= 1/(sqrt{x+h}+ sqrt{h}).

Last edited: Dec 30, 2021
Country Boy, Dec 30, 2021
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3. ### nycmathguy

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I can't wait to start Chapter 1 in my James Stewart Calculus textbook. I am still a few months away from that self-study. I first want to complete my Ron Larson Precalculus review. By the way, I got an A minus in Precalculus (MA172) at Lehman College in the Spring 1993 semester. Boy, I am getting old.

nycmathguy, Dec 31, 2021
4. ### mmesford

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Oh, of course! I tried doing that but without changing the sign of the sqrt{x}. Brilliant!

mmesford, Dec 31, 2021
5. ### mmesford

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I’m working my way through Thomas/Finney for the second time. Last time I only got as far as integration. But I’m retired now and want to be more rigorous. And get further!
As for being old, I had kids in high school in ‘93. Ha!

mmesford, Dec 31, 2021
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