Evaluating the limit of a function

[moouers]

Homework Statement

I'm not sure how to type "lim x-> 0 out using this forum's math symbols...

lim x→0 [\frac{1}{x\sqrt{1+x}}-\frac{1}{x}]

Homework Equations

The Attempt at a Solution

Honestly, I'm just not seeing how to manipulate the equation to get it to a point where I can find the solution. This isn't graded and isn't even for any class I'm in...I'm just curious. It's been 10 years since I've done anything like this, so I'm trying to get up to speed before I take Calculus again in the fall.

 

[genericusrnme]

Since x is going to zero, I'd try a taylor series about x=0 ;)

 

[moouers]

Haha, well, I'd love to. But since this is in one of the first few chapters in the book, long before Taylor series is introduced, there must be another way and I'd like do it that way first.

 

[genericusrnme]

oh.. :p

In that case, try getting everything into one term, with a common denominator and see what happens

 

[moouers]

I tried that and was just as lost. Maybe I did it wrong, but I got:

lim x→0 [\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}]

If I did that correctly, I still don't know where to go from there.

 

[genericusrnme]

also try multiplying by 1 in a nice way :3

 

[moouers]

Would you mind showing my the first step to take, so I can go on from there? I would appreciate it.

 

[Dick]

I tried that and was just as lost. Maybe I did it wrong, but I got:

lim x→0 [\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}]

If I did that correctly, I still don't know where to go from there.

Now try multiplying numerator and denominator by 1+\sqrt{1+x}. Expand out the numerator.

 

[alanlu]

No, the expansion was done correctly.

Try rationalizing the numerator by multiplying by $\frac{1 + \sqrt{1 + x}}{1 + \sqrt{1 + x}}$

Also, Taylor series are used for approximating functions, not evaluating limits.

Edit: ninjas on this board!

 

[moouers]

Wow, so that was the trick. I carried out the math from there and caclulator/book confirms the answer. Thank you all so much. That was far simpler than it seemed.