Getting a variable out of the denominator

[mmesford]

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.

 

[Country Boy]

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}).

 

[nycmathguy]

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}).

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.

 

[mmesford]

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}).

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

 

[mmesford]

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.

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!

 

[nycmathguy]

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!

You are more than welcome join me on my precalculus review ride. I am currently at the end of Chapter 5 in the Ron Larson Precalculus textbook Edition 10E. You can freely download the book by patiently searching online.

I am not retired. I hope to someday be. I can only dedicate time to math on my days off. So, look for my post in the Geometry and Trigonometry forum here. I am lucky to have MathLover1 helping along the way. She is a true math professional.

 

[HallsofIvy]

Every time I complain about being old, my friend, who is older than I am, reminds me or the alternative!

 

[conway]

Every time I complain about being old, my friend, who is older than I am, reminds me or the alternative!

The alternative is better.

Given the "law of conservation of energy". We therefore cannot become "nothing".

Given the "law of love is the greatest of all things".

Given the "law nothing last forever....entropy....decay....."

The alternative "death" is better.

Lastly consider life on an infinite time line...it leaves you bored of all things.

 

[HallsofIvy]

The alternative to getting old is DIEING!

 

[conway]

Obviously....

Death is better than decaying away...

Death is better than becoming immune to happiness...which is what happens with an infinite lifespan...

Death is better than never experiencing a new life....

Imagine your mind riddled with dementia. Imagine your body racked with incurable cancer from limb to limb. From bone to skin. Imagine you breaths labored, and full of pain. Each, and every one. Imagine your fingers stiff with arthritis, unable to bend. I will assume you have mountains of courage, and fortitude. Yet even you would reach a point where you will agree death is better. Death is a blessing from God. It is as much so, as beautiful LIFE, is a blessing from God. There is no light, without the dark.

 

[conway]

Well HallsofIvy at least I have one post you seem to like. You have viewed it enough times. In any case I have decided to retreat from this forum, and mathematics. Better I stick to philosophy hey! Yet I've said this before. It has actually been several years since I've been on line "pushing" my idea. I will try to do better in "quitting" this time. Sincerely I apologize for wasting yours, and the others time.

I hope you are blessed, by God, luck, or accident. Better yet all three!

 

[ROBERTMILLS]

To find the limit of (sqrt{x+h}- sqrt{x})/h as h approaches 0, you can rationalize the numerator. This is done by multiplying both the numerator and denominator by sqrt{x+h}+ sqrt{x}.

This operation yields ((sqrt{x+h}- sqrt{x})(sqrt{x+h}+sqrt{x}))/h(sqrt{x+h}+ sqrt{x}), which simplifies to (x+h- x)/(h(sqrt{x+h}+ sqrt{x})), which further simplifies to h/(h(sqrt{x+h}+ sqrt{x})), and ultimately, 1/(sqrt{x+h}+ sqrt{x}).

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