Limit of a rational function with a square-rooted expression in numerator.

In summary, the conversation discusses different methods for evaluating the limit as x approaches 0 of [(x+4)^1/2-2]/x. The person has tried dividing out by (x+4)^1/2 and by x, but both methods resulted in infinity. The other person suggests using l'Hospital's rule or rationalizing the numerator, and the conversation ends with the acknowledgement that both methods can be used for indeterminate forms.
  • #1
MurdocJensen
47
0
lim as x -> 0, [(x+4)1/2-2]/x

That's the limit I want to evaluate. I keep running into problems getting to the real limit (1/4).
You don't have to give me the answer, but let me know if I'm missing something simple. Or you can just give me a hint.
 
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  • #2
What have you tried so far?
 
  • #3
So far I have tried dividing out by (x+4)1/2. This still gives me an x in the denominator that yields infinity when x->0.

I have also tried dividing out by x, but this gives me fractions in the numerator that, again, give me infiinity.
 
  • #4
Have you learned l'hopital's rule yet?
 
  • #5
Yea, but I was able to get the answer by just rationalizing the numerator. I'm going to try l'Hospital now.

EDIT: I thought we only use l'Hospital's rule for lmits that are 0/0 or inf/inf.
 
  • #6
Both methods should work. l'hopital's rule will probably be easier, but rationalizing the numerator is good practice.
 
  • #7
But aren't we using l'Hospital's rule for indeterminate forms?
 
  • #8
MurdocJensen said:
But aren't we using l'Hospital's rule for indeterminate forms?

You can use l'hopital's rule for 0/0 or +-inf/inf (which in this case you get 0/0). I was just saying there's more than one way of getting the right answer.
 
  • #9
gb7: Yea, I'm an idiot for not noticing that. Thanks for the help!
 

1. What is a rational function with a square-rooted expression in the numerator?

A rational function is a mathematical function that can be expressed as a ratio of two polynomial functions. A square-rooted expression in the numerator is a function where the expression inside the square root symbol is in the numerator of the rational function.

2. How do you find the limit of a rational function with a square-rooted expression in the numerator?

To find the limit of a rational function with a square-rooted expression in the numerator, you can use the same techniques as finding the limit of any other rational function. This can include simplifying the function, factoring the expression, and using limit rules such as the Squeeze Theorem or L'Hopital's Rule.

3. What is the purpose of finding the limit of a rational function with a square-rooted expression in the numerator?

The limit of a rational function with a square-rooted expression in the numerator can help us understand the behavior of the function as it approaches a certain value. This can be useful in applications such as finding the maximum or minimum value of a function or determining the continuity of a function.

4. Are there any special cases when finding the limit of a rational function with a square-rooted expression in the numerator?

Yes, there are some special cases to consider when finding the limit of a rational function with a square-rooted expression in the numerator. For example, if the numerator and denominator both approach zero, we may need to use more advanced techniques such as the Squeeze Theorem or L'Hopital's Rule. Additionally, if the degree of the numerator is greater than the degree of the denominator, the limit may approach positive or negative infinity.

5. Can you provide an example of finding the limit of a rational function with a square-rooted expression in the numerator?

Sure, let's consider the function f(x) = (x^2 - 4)/(√x - 2). To find the limit as x approaches 4, we can simplify the function by factoring the numerator and denominator. This gives us f(x) = (x+2)(x-2)/(√x - 2)(√x + 2). We can then cancel out the common factor of (x-2) and use the limit rule that states that the limit of a quotient is equal to the quotient of the limits. Therefore, the limit of f(x) as x approaches 4 is equal to 4.

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