Find the limit of the following function

[Ryker]

Homework Statement

\lim_{x \rightarrow 1} \frac{\sqrt{3+x} - 2}{\sqrt{4x} - 2}

Homework Equations

The Attempt at a Solution

I'm completely lost on this one. I've tried approaching it by taking a sequence that converges to 1 and plugging that value into the equation, but then I stumble upon the denominator being 0. I would be really grateful if anyone could show me how to do these types of problems, as I've got a midterm in a couple of hours. How exactly do I get rid of that 0 to be able to do something meaningful with the expression?

Thanks in advance.

 

[sjb-2812]

One way, perhaps not the best is to consider the value of the function at x=1.1? at x=1.01? at x=1.001? And at 0.9? at 0.99? at 0.999?

 

[Ryker]

Yeah, I mean, the thing is I can find online graph drawing tools and see what the limit is, but I would like to know how to prove it. And I'd also like to do these problems when I don't have a computer or calculator at hand. So supposing something like this gets thrown onto a midterm or final exam, how would one approach it? I'm trying to somehow factor out something, but am having no luck thus far.

 

[dreit]

This yields an indeterminate form, 0/0, so you can use L'hopitals rule to find this answer.

 

[dreit]

If you don't know this calculus rule, try to rationalize the denominator which should give you the same answer

 

[Ryker]

We haven't done L'hopitals rule yet, so I don't really know how to use it. And as far as rationalization is concerned, I'm trying to do that, but I'm still unable to get rid of 0 in the denominator. What should I multiply the expression with?

 

[Ryker]

Alright, so I got it with the help of https://www.physicsforums.com/showthread.php?t=248889" topic Thanks for the your suggestions, guys, as well.

 

[SEngstrom]

A simple way (to remember how to do this) is to look at a series expansion of the numerator and the denominator.

 

[Ryker]

Yeah, I think I'm going to remember this strategy from now on, the only problem was that we haven't done these hard limit examples in class yet, but only the simplest ones. Now I was aware of denominator rationalization from high school, but not with simultaneous denominator and numerator rationalization. And not seeing this before, I found it hard to really come up with that tactic on my own.

Although, come to think of it, what exactly do you mean by series expansion? I'm not from an English-speaking country originally, so I might know what it is, but am just not familiar with the terminology. Could you give an example?

edit: I also hate how frustrated with myself I get when I try to solve stuff like that which I have no clue how to approach, but then when I find out how to do it, I start enjoying it immensely

 

[micromass]

What if you multiply the numerator and the denumerator by \sqrt{4x}-2.
Then multiply the numerator and denumerator by \sqrt{3+x}-2. I think this should get you somewhere...

 

[SEngstrom]

Expand numerator and denominator as a power (Taylor) series around x=1
\sqrt{3+x}-2={{x-1}\over 4}-{1\over 64}(x-1)^2+higher order terms
\sqrt{4x}-2=(x-1)-{1\over 4}(x-1)^2+higher order terms

The linear terms makes the behavior around x=1 obvious.

 

[Ryker]

What if you multiply the numerator and the denumerator by \sqrt{4x}-2.
Then multiply the numerator and denumerator by \sqrt{3+x}-2. I think this should get you somewhere...

Yeah, that's what I did (except probably you mistyped and meant the conjugate of those expressions, right?), and it worked perfectly.

 

[Ryker]

Expand numerator and denominator as a power (Taylor) series around x=1
\sqrt{3+x}-2={{x-1}\over 4}-{1\over 64}(x-1)^2+higher order terms
\sqrt{4x}-2=(x-1)-{1\over 4}(x-1)^2+higher order terms

The linear terms makes the behavior around x=1 obvious.

Thanks for the explanation, I'll try and look into that. This is only my first semester at a university, though, so we haven't stumbled upon Taylor series yet.

edit: I've looked at the curriculum, and it seems we're doing that next year. Are we going slow, and are Taylor series usually introduced in the first term of the first year usually?

 

[SEngstrom]

No I'm sure your curriculum is fine :-) It is just a tool that happens to make sense to me for understanding (practical) limit behavior like in this problem...

 

[╔(σ_σ)╝]

Rationalizing is the usual trick for limits involving square roots also forget the stuff about Taylor series it is hardly useful in computing limits.

 

[Dick]

What's the limit x->1 of (sqrt(3+x)+2)/(sqrt(4x)+2)? Just multiply by that expression.

 

[Ryker]

Thanks again, guys, I've got it nailed down now