Find the limit of the following function

In summary, the student is struggling with a homework problem that asks for the limit as x approaches 1 of the quotient of the square root of 3+x and the square root of 4x. The student finds a power series solution around x=1, and then uses L'hopitals rule to find the answer.
  • #1
Ryker
1,086
2

Homework Statement


[tex]\lim_{x \rightarrow 1} \frac{\sqrt{3+x} - 2}{\sqrt{4x} - 2}[/tex]

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.
 
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  • #2
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?
 
  • #3
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.
 
  • #4
This yields an indeterminate form, 0/0, so you can use L'hopitals rule to find this answer.
 
  • #5
If you don't know this calculus rule, try to rationalize the denominator which should give you the same answer
 
  • #6
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?
 
  • #8
A simple way (to remember how to do this) is to look at a series expansion of the numerator and the denominator.
 
  • #9
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? :smile: 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 :grumpy:
 
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  • #10
What if you multiply the numerator and the denumerator by [tex]\sqrt{4x}-2[/tex].
Then multiply the numerator and denumerator by [tex]\sqrt{3+x}-2[/tex]. I think this should get you somewhere...
 
  • #11
Expand numerator and denominator as a power (Taylor) series around x=1
[tex]\sqrt{3+x}-2={{x-1}\over 4}-{1\over 64}(x-1)^2+[/tex]higher order terms
[tex]\sqrt{4x}-2=(x-1)-{1\over 4}(x-1)^2+[/tex]higher order terms

The linear terms makes the behavior around x=1 obvious.
 
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  • #12
micromass said:
What if you multiply the numerator and the denumerator by [tex]\sqrt{4x}-2[/tex].
Then multiply the numerator and denumerator by [tex]\sqrt{3+x}-2[/tex]. 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.
 
  • #13
SEngstrom said:
Expand numerator and denominator as a power (Taylor) series around x=1
[tex]\sqrt{3+x}-2={{x-1}\over 4}-{1\over 64}(x-1)^2+[/tex]higher order terms
[tex]\sqrt{4x}-2=(x-1)-{1\over 4}(x-1)^2+[/tex]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?
 
  • #14
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...
 
  • #15
Rationalizing is the usual trick for limits involving square roots also forget the stuff about Taylor series it is hardly useful in computing limits.
 
  • #16
What's the limit x->1 of (sqrt(3+x)+2)/(sqrt(4x)+2)? Just multiply by that expression.
 
  • #17
Thanks again, guys, I've got it nailed down now :biggrin:
 

1. What is a limit in calculus?

A limit in calculus is the value that a function approaches as the input approaches a certain value. It represents the behavior of the function at a specific point and can help determine the continuity and differentiability of the function.

2. How do you find the limit of a function?

To find the limit of a function, you can use various methods such as direct substitution, factoring, and rationalizing the numerator or denominator. You can also use algebraic manipulation or graphing techniques to help determine the limit.

3. What happens if the limit of a function does not exist?

If the limit of a function does not exist, it means that the function does not approach a specific value as the input approaches a certain value. This could be due to a jump or discontinuity in the function, or the function may approach different values from different directions.

4. Can a function have a limit at a point but not be defined at that point?

Yes, a function can have a limit at a point even if it is not defined at that point. This means that the function may have a value that it approaches at that point, but it is not defined or continuous at that specific point.

5. How can limits be used in real-life applications?

Limits are used in various real-life applications, such as in physics, engineering, and economics, to model and analyze the behavior of systems or functions. For example, limits can be used to determine the maximum capacity of a bridge or the optimal production level for a company.

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