Evaluate the limit of [(6-x)^0.5 - 2]/[(3-x)^0.5 - 1]

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In summary, the conversation is discussing ways to evaluate the limit [(6-x)^0.5 - 2]/[(3-x)^0.5 - 1] as x approaches 2. The participants have tried using rationalization and the squeeze theorem, but have not found a solution. One participant suggests using L'Hospital's rule, but the other has not learned it yet. Another suggests using the numerical approach, but notes that it can be tedious. The conversation ends with a hint to rationalize twice, first in the numerator and then in the denominator, to eventually apply direct substitution.
  • #1
MathewsMD
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Evaluate:

lim [(6-x)^0.5 - 2]/[(3-x)^0.5 - 1]
x→2

I have tried rationalizing, but doing so does not help.

I have also tried using the squeeze theorem to solve the question, but I have not found any values that can be used to precisely find the limit. The values I could think of were -x^2 and and x^2, but that does not allow you to find the exact answer.

Any help or tips with this question would be greatly appreciated!
 
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  • #2
Have you tried L'Hospital's rule?
 
  • #3
I just started introductory calculus and have not learned it yet. Any help with or without using L'Hospital's rule would be great!
 
  • #4
I am pretty sure that it is going to be impossible for you to evaluate this limit analytically without the use of L'Hospital's rule. May I ask exactly what level of math you are in and what chapter? I am thinking they might want you to do this using the numerical approach. Which is to say the least, very tedious.
 
  • #5
Multiply the top and bottom by (1/(3-x)^.5) I think that'll do it.
 
  • #6
I am currently on Chapter 2 in Stewart's Calculus (Calculus I). I've learned Squeeze Theorem and other basic limit solving strategies, but nothing too advanced.

And I believe I have tried rationalizing using that root, but have not gotten a solution.
 
  • #7
This may be a big hint, but you will have to rationalize twice. First, rationalize the numerator. Simplify the numerator, but don't multiply it out in the denominator. Then rationalize the denominator. Here, you will need to multiply out two of the factors in the denominator. Eventually, you'll be able to apply direct substitution.
 
  • #8
By rationalize do you mean multiply the conjugate? I don't see how you can rationalize (multiply top and bottom by the root), but you say you only do it to the top and then only do it to the bottom. I don't think you can do that. Also if I multiply the top conjugate and then bottom conjugate it gets REALLY messy and it can't be right.
 
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1. What is the limit of the given expression as x approaches 3?

The limit of the expression is undefined as it results in a division by zero when x approaches 3.

2. Can the limit be evaluated by direct substitution?

No, direct substitution is not possible as it results in a division by zero.

3. How can we evaluate the limit algebraically?

We can use algebraic manipulation and factoring to simplify the expression and then apply the limit rule for square roots to evaluate the limit.

4. What is the behavior of the expression as x approaches positive or negative infinity?

As x approaches positive infinity, the expression approaches 0. As x approaches negative infinity, the expression approaches 2.

5. Can we use a graph to determine the limit?

Yes, we can use a graph to estimate the limit by looking at the behavior of the expression as x approaches the limit point.

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