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Given: $\displaystyle \lim_{x\rightarrow \infty} x\left(\frac{1}{\sqrt{x}} + \sqrt{x+1} \right)=\frac{1}{2}$

why is it $\frac{1}{2}$?? i keep getting infinity

this is what i did

1st. $\sqrt{x+1}-\sqrt{x}= \frac{1}{\sqrt{x}}+\sqrt{x+1}$

2nd. $\lim_{x \to \infty}$ $\sqrt{x}$ over $\frac{1}{\sqrt{x}+\sqrt{x+1}}$

3rd. $\lim_{x \to \infty}\frac{\sqrt{x}}{\sqrt{x}} + \frac{1}{\sqrt{x+1}}$

4th. $1+0=1$

i do not understand how it is $\frac{1}{2}$, not $1$?

any help would be great

thank you

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There would likely be much quicker answers if the question was more readable. – André Nicolas Mar 23 '14 at 19:22
Or if the question was just readable....what's that "34th.1" in the 5th line, for example?? – DonAntonio Mar 23 '14 at 19:23
I don't understand the first step. Also, you have a mistake after the third step. Note that $\frac{a}{b+c}$ is not $\frac{a}{b}+\frac{1}{c}$... – Ludolila Mar 23 '14 at 19:52
@Ludolila, you and Samy (read answers below) decyphered the question in two different ways...Who are we to believe? That's why it is not a good idea to answer/comment on questions that are not clearly written... – DonAntonio Mar 23 '14 at 20:17

$$\require{cancel}\lim_{x\to\infty}\frac{ \sqrt x}{\sqrt x+\sqrt{x+1}}=\lim_{x\to\infty}\frac {\cancel{\sqrt x}}{\cancel{\sqrt x}\left(1+\sqrt{1+\frac1x}\right)}=\lim_{x\to\infty}\frac {1}{\left(1+\sqrt{1+\frac1x}\right)}=\frac12$$

Edit $$\lim_{x\to\infty}\underbrace{x}_{\to\infty}\left(\underbrace{\frac{1}{\sqrt{x}} + \sqrt{x+1}}_{\to\infty} \right)=+\infty$$

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this helps much. thank you so much. and to those of you who answered, i am so sorry for confusion and i don't know how to use LaTex. again6 so sorry. my original question was this equation. – chris Mar 23 '14 at 20:30
@chris I edited my answer. – user63181 Mar 23 '14 at 20:41
what you wrote originally is the right question i meant to ask. – chris Mar 24 '14 at 2:32
thank you so much though. i wasn't sure how to get 1/2 but i do know. you've shown the correct work :) – chris Mar 24 '14 at 2:33
You're welcome. – user63181 Mar 24 '14 at 6:15

You have: x^(1/2)/(x^(1/2) + (x+1)^(1/2)) = 1/(1 + (1 + 1/x)^(1/2)) ---> 1/(1 + 1) = 1/2 as x --> infinity.

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answering without using LaTeX formating automatically lowers your chances of being read and, eventually, upvoted by at least 1/50+48%...approx. BTW, what number did I write? – DonAntonio Mar 23 '14 at 20:19

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