How can rationalizing the numerator help solve limits algebraically?

[famallama]

Homework Statement

Evaluate limit as x approaches 0 of (square root(4+x^4)-2)/x^4) algebraically by rationalizing the numerator. Show details

The Attempt at a Solution

I rationalized the numerator and i see it as there is a root in the denominator now which is when i was taught to rationalize

 

[HallsofIvy]

Some time ago, we had a question on this forum basically asking why you always rationalize the numerator! The answer is, of course, that you don't always- although for many basic algebra problems, such as adding fractions, that helps. I have seen texts that devote quite a lot of time to rationalizing the numerator as well.

Okay,if you have rationalized the numerator, you will have a square root in the denominator- but that doesn't hurt. It should be of the form \sqrt{4+ x^4}+ 2 which goes to 4, not 0, as x goes to 2. What happens to the rest of the fraction? That's the important thing!

 

[famallama]

that doesn't really help. I get x^4 -4(sqrt(4+x^4)) +6 on top of x^4(sqrt(4+x^))-2x^4

 

[tiny-tim]

Hi famallama!

(have a square-root: √ )

that doesn't really help. I get x^4 -4(sqrt(4+x^4)) +6 on top of x^4(sqrt(4+x^))-2x^4

erm … the object is to have no √ on the top

Hint: your factors had a - on the top and the bottom … try it with a +

 

[HallsofIvy]

that doesn't really help. I get x^4 -4(sqrt(4+x^4)) +6 on top of x^4(sqrt(4+x^))-2x^4

Well, you shouldn't. Since "rationalizing" the numerator should get rid of the square root in the numerator, it appears you haven't done that properly. If you would show your work we might be able to clarify things.