Evaluate the intermediate difference with L'Hospital's rule

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In summary, the problem involves finding the limit of (x-sqrt(x^2 -1)) as x approaches infinity. The attempted solution involves using L'Hospital's rule, which can be spelled as L'hospital or L'Hôpital, to eliminate the radical and factor out an x. Wikipedia notes that the pronunciation can vary, but the correct pronunciation is [lopiˈtal].
  • #1
Painguy
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Homework Statement



limit (x-sqrt(x^2 -1)) x-> inf

Homework Equations





The Attempt at a Solution



What I had in mind was to somehow get rid of the radical so I can factor out an x then use L'Hospital's rule with the resulting intermediate product. I'm not sure how to approach the problem however.
 
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  • #2
L'hopital, not L'hospital. LOL.
 
  • #3
operationsres said:
L'hopital, not L'hospital. LOL.

http://mathworld.wolfram.com/LHospitalsRule.html

My book spells it as L'hospital, and so does Wolfram Alpha.

Wikipedia says, "In calculus, l'Hôpital's rule pronounced: [lopiˈtal] (also sometimes spelled l'Hospital's rule with silent "s" and identical pronunciation)"
 
  • #4
Nice!
 

1. What is L'Hospital's rule?

L'Hospital's rule is a mathematical tool used to evaluate the limit of a function that is in an indeterminate form, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions is in an indeterminate form, then the limit of the ratio of their derivatives will be equal to the original limit.

2. When should L'Hospital's rule be used?

L'Hospital's rule should only be used when the limit of a function is in an indeterminate form and all standard methods of limit evaluation, such as direct substitution, do not work. It is important to check if the conditions for using L'Hospital's rule are met before applying it.

3. How do you apply L'Hospital's rule?

To apply L'Hospital's rule, take the derivative of both the numerator and denominator of the original function. Then, evaluate the limit of the ratio of the derivatives. If the limit still remains in an indeterminate form, continue taking derivatives until the limit can be evaluated or until the derivative becomes constant.

4. What are the conditions for using L'Hospital's rule?

The conditions for using L'Hospital's rule are that the limit must be in an indeterminate form (0/0 or ∞/∞), the functions must be differentiable in a neighborhood of the limit point, and the limit of the ratio of the derivatives must exist.

5. Are there any limitations to L'Hospital's rule?

Yes, there are some limitations to L'Hospital's rule. It cannot be used to evaluate limits at infinity, and it may not always give the correct result if used incorrectly or on the wrong type of function. It is important to check the conditions and use it correctly to ensure accurate results.

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