What is the limit of a complex fraction with L'Hopital's rule?

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In summary, "S8.1.6.64 lim w/ L'H" refers to the limit with L'Hopital's rule, a mathematical concept used to find the limit of a function as it approaches a certain value. L'Hopital's rule can be used when both the numerator and denominator of a fraction approach 0 or infinity, and it is most commonly used for limits that result in an indeterminate form. However, there are limitations to using this rule and it is important to check if the conditions are met and consider other methods if necessary. To practice solving limits with L'Hopital's rule, there are many online resources available such as practice problems and tutorials.
  • #1
karush
Gold Member
MHB
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5
\tiny{s8.1.6.64}
Evaluate
$\displaystyle\lim_{x \to 2}
\dfrac{\sqrt{6-x}{-2}}{\sqrt{3-x}-1}$

ok so if you plug in 2 directly you get $\dfrac{0}{0}$

So we either use L'H rule or use conjugate

or is there better way
 
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  • #2
L'Hopital works

you can also multiply numerator and denominator by both conjugates
 
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FAQ: What is the limit of a complex fraction with L'Hopital's rule?

What is L'Hopital's rule?

L'Hopital's rule is a mathematical theorem used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions f(x) and g(x) is indeterminate, then the limit of the ratio of their derivatives f'(x) and g'(x) will be the same as the original limit.

How does L'Hopital's rule apply to complex fractions?

L'Hopital's rule can be applied to complex fractions by breaking them down into simpler fractions and then using the rule to evaluate the limit. This can be done by finding the common denominator and then taking the derivative of the numerator and denominator separately.

What is the process for using L'Hopital's rule on a complex fraction?

The process for using L'Hopital's rule on a complex fraction is as follows:

  1. Simplify the complex fraction by finding the common denominator.
  2. Take the derivative of the numerator and denominator separately.
  3. Evaluate the limit of the new fraction.
  4. If the limit is still indeterminate, repeat the process until a value is obtained or it is determined that the limit does not exist.

What are the limitations of using L'Hopital's rule on complex fractions?

L'Hopital's rule can only be used on indeterminate forms of 0/0 or ∞/∞. It cannot be used on other types of indeterminate forms, such as 1^∞ or ∞ - ∞. Additionally, the rule may not always provide a definitive answer and further methods may be needed to evaluate the limit.

Can L'Hopital's rule be used on limits that approach infinity?

Yes, L'Hopital's rule can be used on limits that approach infinity as long as the limit is in the form of ∞/∞. In this case, the rule can be applied by taking the derivatives of the numerator and denominator separately and evaluating the limit of the resulting fraction.

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