What is the Limit of (4[(SQRT(x+2)) – (SQRT2))]/x as x approaches 0?

[ladyrae]

Help! Limit Problem

Find the limit by analytic methods:

lim x->0 (4[(SQRT(x+2)) – (SQRT2))]/x

The first part of the problem asked me to estimate the limit by using a table and I came up with 1.414.

I tried using the limit laws and came up with SQRT((4)(2))/(0+ SQRT((4)(2)))=1

Any suggestions?

 

[Math Is Hard]

Did you try multiplying a conjugate? That would be my first instinct.
Actually, my very first instinct would be to use L'Hospital's rule but I am guessing y'all haven't done that yet...?

 

[master_coda]

L'Hôpital's rule works for this limit.

EDIT: It seems that someone else answered before me...I knew I shouldn't have wasted time trying to remember how to type an "o" with a circumflex.

 

[TALewis]

L'Hôpital's rule should work, but it's still possible to find the limit with algebraic methods (and that might be how ladyrae is expected to do it).

We want to find:

$$\lim_{x\rightarrow 0} \frac{4(\sqrt{x+2} - \sqrt{2})}{x}$$

I agree with Math Is Hard: I would multiply the top and bottom by the numerator's conjugate, like this:

$$\left(\frac{4(\sqrt{x+2} - \sqrt{2})}{x}\right)<br /> \left(\frac{\sqrt{x+2} + \sqrt{2}}{\sqrt{x+2} + \sqrt{2}}\right)$$

Expanding that out, you should be able to plug in zero and find the limit directly.

Also note that your analytic result should be close to your table result (what you get from plugging in values exceedingly close to zero, like 0.000001). If you get 1.414 numerically and 1 analytically, you've probably made a mistake.

 

[Math Is Hard]

MC: Your circumflex "o" looks very elegant however. I think it was worth the trouble!