((x+1)^0.5+(x+2)^0.5-(4*x+12)^0.5)/((x)^0.5+(x+2)^0.5-(4*x+12)^0.5)

[Neyestan]

I need step by step solution of finding the limit when x goes to +inf.

((x+1)^0.5+(x+2)^0.5-(4*x+12)^0.5)/((x)^0.5+(x+2)^0.5-(4*x+12)^0.5)

Thank you very much




I am sorry that i posted 3 times, I am new in here

 

[Neyestan]

Limit of x --> inf.

I have difficulties solving this problem, and I need step by step solution please:

((x+1)^0.5+(x+2)^0.5-(4*x+12)^0.5)/((x)^0.5+(x+2)^0.5-(4*x+12)^0.5)

when x ---> inf.

Thank you very much



I am sorry that I posted 3 times, I am new in here :smile:

 

[Neyestan]

Limit problem

I am sorry that I posted 3 times, I am new in here :smile:

 

[stapel]

I need step by step solution of finding the limit when x goes to +inf.

You already have loads of step-by-step solutions in your book, in your class notes, in the examples on the various web sites you've researched, etc, etc. One more fully-worked example (which we don't provide here anyway) is unlikely to fix whatever is the problem. So instead, let's try working in accordance to the policy you saw in the "Read Before Posting" thread that you read before posting.

((x+1)^0.5+(x+2)^0.5-(4*x+12)^0.5)/((x)^0.5+(x+2)^0.5-(4*x+12)^0.5)

Nest parentheses, etc, can sometimes be hard to decipher. Below is what I think you mean by the above:

. . . . .\(\displaystyle \dfrac{\sqrt{\strut x\, +\, 1\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12}}{\sqrt{\strut x\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12\,}}\)

Is this correct? If not, please reply with corrections and clarification. If so, please reply with your thoughts and efforts so far. For instance, did you start by dividing, top and bottom, by the square root of x? And so forth. Thank you!

 

[stapel]

I am sorry that I posted 3 times, I am new in here :smile:

Okay. Well, now I know that I got your meaning correct. (I'd replied to one of the posts without the image.) I also provided you with a link to the forum's policy; namely, the "Read Before Posting" thread.

Kindly please review that thread, and then reply with your thoughts and efforts so far. Thank you!

 

[Neyestan]

Okay. Well, now I know that I got your meaning correct. (I'd replied to one of the posts without the image.) I also provided you with a link to the forum's policy; namely, the "Read Before Posting" thread.

Kindly please review that thread, and then reply with your thoughts and efforts so far. Thank you!

Thank you very much for your response, I actually think that this problem is either very tricky or the solution is right in front of my eyes and I don't see it
I tried to find the limit by factoring, by conjugate and LCD but none of them worked, and I also tried to divide top and bottom by square root of x but I am stuck.[h=2]I really appreciate your help[/h]

 

[Neyestan]

Okay. Well, now I know that I got your meaning correct. (I'd replied to one of the posts without the image.) I also provided you with a link to the forum's policy; namely, the "Read Before Posting" thread.

Kindly please review that thread, and then reply with your thoughts and efforts so far. Thank you!

 

[stapel]

This is what I got after dividing the top and the bottom by square root of x.

It is ugly.

. . . . .\(\displaystyle \dfrac{ \dfrac{1}{\sqrt{\strut \dfrac{x}{x\, +\, 1}\,}}\, +\, \dfrac{1}{\sqrt{\strut \dfrac{x}{x\, +\, 2}\,}}\, -\, \dfrac{2}{\sqrt{\strut \dfrac{x}{x\, +\, 3}\,}} }{1\, +\, \dfrac{1}{\sqrt{\strut \dfrac{x}{x\, +\, 2}\,}}\, -\, \dfrac{2}{\sqrt{\strut \dfrac{x}{x\, +\, 3}\,}} }\)

How did you arrive at this? What were your steps?

For instance, I would have started like this:

. . . . .\(\displaystyle \mbox{a. }\, \dfrac{\sqrt{\strut x\, +\, 1\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12}}{\sqrt{\strut x\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12\,}}\)

. . . . .\(\displaystyle \mbox{b. }\, \dfrac{\left(\sqrt{\strut x\, +\, 1\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12}\right)\, \cdot\, \left(\dfrac{1}{\sqrt{\strut x\,}}\right) }{ \left(\sqrt{\strut x\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12\,}\right)\, \cdot\, \left(\dfrac{1}{\sqrt{\strut x\,}}\right)}\)

. . . . .\(\displaystyle \mbox{c. }\, \dfrac{ \dfrac{\sqrt{\strut x\, +\, 1\,}}{\sqrt{\strut x\,}}\, +\, \dfrac{\sqrt{\strut x\, +\, 2\,}}{\sqrt{\strut x\,}}\, -\, \dfrac{\sqrt{\strut 4x\, +\, 12}}{\sqrt{\strut x\,}} }{ \dfrac{\sqrt{\strut x\,}}{\sqrt{\strut x\,}}\, +\, \dfrac{\sqrt{\strut x\, +\, 2\,}}{\sqrt{\strut x\,}}\, -\, \dfrac{\sqrt{\strut 4x\, +\, 12\,}}{\sqrt{\strut x\,}} }\)

. . . . .\(\displaystyle \mbox{d. }\, \dfrac{ \sqrt{\strut \dfrac{x\, +\, 1}{x}\,}\, +\, \sqrt{\strut \dfrac{x\, +\, 2}{x}\,}\, -\, \sqrt{\strut \dfrac{4x\, +\, 12}{x}\, } }{ \sqrt{\strut \dfrac{x}{x}\,}\, +\, \sqrt{\strut \dfrac{x\, +\, 2}{x}\,}\, -\, \sqrt{\strut \dfrac{4x\, +\, 12}{x}\, } }\)

. . . . .\(\displaystyle \mbox{e. }\, \dfrac{ \sqrt{\strut 1\, +\, \dfrac{1}{x}\,}\, +\, \sqrt{\strut 1\, +\, \dfrac{2}{x}\,}\, -\, \sqrt{\strut 4\, +\, \dfrac{12}{x}\,} }{ \sqrt{\strut 1\,}\, +\, \sqrt{\strut 1\, +\, \dfrac{2}{x}\,}\, -\, \sqrt{\strut 4\, +\, \dfrac{12}{x}\,} }\)

(The steps are named, for ease of reference.)

What did you do? Please be complete. Thank you!

 

[Ishuda]

Why do you write it that way? Why not as
\(\displaystyle \frac{\sqrt{1 +\, \frac{1}{x}}\, + \sqrt{1 +\, \frac{2}{x}}\, -\, 2\, \sqrt{1\, +\, \frac{3}{x}}}{1\, +\, \sqrt{1 +\, \frac{2}{x}}\, -\, 2\, \sqrt{1 +\, \frac{3}{x}}}\)

 

[Neyestan]

How did you arrive at this? What were your steps?

For instance, I would have started like this:

. . . . .\(\displaystyle \mbox{a. }\, \dfrac{\sqrt{\strut x\, +\, 1\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12}}{\sqrt{\strut x\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12\,}}\)

. . . . .\(\displaystyle \mbox{b. }\, \dfrac{\left(\sqrt{\strut x\, +\, 1\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12}\right)\, \cdot\, \left(\dfrac{1}{\sqrt{\strut x\,}}\right) }{ \left(\sqrt{\strut x\,}\, +\, \sqrt{\strut x\, +\, 2\,}\, -\, \sqrt{\strut 4x\, +\, 12\,}\right)\, \cdot\, \left(\dfrac{1}{\sqrt{\strut x\,}}\right)}\)

. . . . .\(\displaystyle \mbox{c. }\, \dfrac{ \dfrac{\sqrt{\strut x\, +\, 1\,}}{\sqrt{\strut x\,}}\, +\, \dfrac{\sqrt{\strut x\, +\, 2\,}}{\sqrt{\strut x\,}}\, -\, \dfrac{\sqrt{\strut 4x\, +\, 12}}{\sqrt{\strut x\,}} }{ \dfrac{\sqrt{\strut x\,}}{\sqrt{\strut x\,}}\, +\, \dfrac{\sqrt{\strut x\, +\, 2\,}}{\sqrt{\strut x\,}}\, -\, \dfrac{\sqrt{\strut 4x\, +\, 12\,}}{\sqrt{\strut x\,}} }\)

. . . . .\(\displaystyle \mbox{d. }\, \dfrac{ \sqrt{\strut \dfrac{x\, +\, 1}{x}\,}\, +\, \sqrt{\strut \dfrac{x\, +\, 2}{x}\,}\, -\, \sqrt{\strut \dfrac{4x\, +\, 12}{x}\, } }{ \sqrt{\strut \dfrac{x}{x}\,}\, +\, \sqrt{\strut \dfrac{x\, +\, 2}{x}\,}\, -\, \sqrt{\strut \dfrac{4x\, +\, 12}{x}\, } }\)

. . . . .\(\displaystyle \mbox{e. }\, \dfrac{ \sqrt{\strut 1\, +\, \dfrac{1}{x}\,}\, +\, \sqrt{\strut 1\, +\, \dfrac{2}{x}\,}\, -\, \sqrt{\strut 4\, +\, \dfrac{12}{x}\,} }{ \sqrt{\strut 1\,}\, +\, \sqrt{\strut 1\, +\, \dfrac{2}{x}\,}\, -\, \sqrt{\strut 4\, +\, \dfrac{12}{x}\,} }\)

(The steps are named, for ease of reference.)

What did you do? Please be complete. Thank you!

Yes you are right; thanks.
I did exactly as your steps but somehow in step "d" I made a mistake and became upside down.
And I also used a calculator and the answer was 0.75!!

 

[Neyestan]

Why do you write it that way? Why not as
\(\displaystyle \frac{\sqrt{1 +\, \frac{1}{x}}\, + \sqrt{1 +\, \frac{2}{x}}\, -\, 2\, \sqrt{1\, +\, \frac{3}{x}}}{1\, +\, \sqrt{1 +\, \frac{2}{x}}\, -\, 2\, \sqrt{1 +\, \frac{3}{x}}}\)

Thank you Ishuda, you are right and I made a mistake, I am working on this problem for 3 days and my mind is not responding well.
Stapel got the same thing as you got but still if I substitute the infinity, the numerator become zero.

 

[Neyestan]

Thanks everybody, I appreciate your time and effort to help me out.
Although I still not able to solve the problem.

 

[stapel]

Thanks everybody, I appreciate your time and effort to help me out.
Although I still not able to solve the problem.

You took the nearly-completed solution you were provided, you took the limit, you got a value, and... then what? Where are you stuck? Please be specific. Thank you!

 

[Neyestan]

You took the nearly-completed solution you were provided, you took the limit, you got a value, and... then what? Where are you stuck? Please be specific. Thank you!

I should thank you and other kind people here, I finally solved it.
My knowledge of math in calculus is very weak and I am trying to improve it.
Again I thank you all folk, you all did a grate job.

 

[Mohammed Hashim]

Plotting the function (Using google search) May help you

 

[Ishuda]

I should thank you and other kind people here, I finally solved it.
My knowledge of math in calculus is very weak and I am trying to improve it.
Again I thank you all folk, you all did a grate job.

What was your solution?

 

[Neyestan]