# Thread: Help Solving a limit

1. ## Help Solving a limit

Hey everyone,

Can anyone help solve this limit? I'm having some difficulty.

$formdata=\lim_{t+\to+0}+\frac1{t\sqrt{1+t}}+-+\frac1+{t}$

Thanks

2. Hello evant8950

Welcome to Math Help Forum!
Originally Posted by evant8950
Hey everyone,

Can anyone help solve this limit? I'm having some difficulty.

$formdata=\lim_{x+->+0}+\frac1{t\sqrt{1+t}}+-+\frac1+{t}$

Thanks
I assume you mean:
$\displaystyle \lim_{x\to0}\left(\frac{1}{x\sqrt{1+x}}-\frac1x\right)$
In which case write it as:
$\displaystyle \frac{1}{x\sqrt{1+x}}-\frac1x =\frac1x\Big((1+x)^{-\frac12}-1\Big)$
Now write the Binomial expansion of $\displaystyle (1+x)^{-\frac12}$, and simplify. Then you'll find that
$\displaystyle \lim_{x\to0}\left(\frac{1}{x\sqrt{1+x}}-\frac1x\right)=-\frac12$

3. Originally Posted by evant8950
Hey everyone,

Can anyone help solve this limit? I'm having some difficulty.

$formdata=\lim_{t+\to+0}+\frac1{t\sqrt{1+t}}+-+\frac1+{t}$

Thanks
For $\displaystyle \lim _{t \to 0} \frac{1 - \sqrt{1+t}}{t \sqrt{1+t}}$ and multiplying by $\displaystyle \frac{1 + \sqrt{1+t}}{1 + \sqrt{1+t}}$ gives

$\displaystyle \lim_{t \to 0} \frac{-1}{(1+\sqrt{1+t})\sqrt{1+t}}$

I think you can finish.

4. Originally Posted by evant8950
Hey everyone,

Can anyone help solve this limit? I'm having some difficulty.

$formdata=\lim_{t+\to+0}+\frac1{t\sqrt{1+t}}+-+\frac1+{t}$

Thanks

$\displaystyle \frac{1}{t\sqrt{1+t}}-\frac{1}{t}=\frac{1-\sqrt{1+t}}{t\sqrt{1+t}}=$ $\displaystyle \frac{-t}{t\sqrt{1+t}\left(1+\sqrt{1+t}\right)}$ $\displaystyle =-\frac{1}{\sqrt{1+t}\left(1+\sqrt{1+t}\right)}\xrig htarrow[t\to 0]{} ...?$

Tonio

Oh, it never mind: somebody already did ALL the work for you. **sigh**

5. Mmm..

We have $\displaystyle \frac{1}{t\sqrt{1+t}}-\frac{1}{t}=\frac{1}{t}\left( \frac{1-\sqrt{1+t}}{\sqrt{1+t}} \right)=-\frac{1}{\left( 1+\sqrt{1+t} \right)\sqrt{1+t}},$ for $\displaystyle t\ne0,$ now as $\displaystyle t\to0$ yields the result.

(Ahhh, I'm slow today.)

6. Thanks everyone for the help. I appreciate it.

7. x=Root(t+1)

so $formdata=\lim_{t+\to+0}+\frac1{t\sqrt{1+t}}+-+\frac1+{t}$=(1-x)/(-x)(x+1)(1-x)=1/(-x^2-x)

t->0 means x->1

lim 1/(-x^2-x)=-0.5
x->1

OK..

8. oops..I came late..