Finding a limit without l'hopial

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In summary, a practice question in a grade 12 mathematics class involves finding the limit as x approaches 2 of ((sqrt(6-x)-2) / (sqrt(3-x) - 1)). The poster is able to easily solve it using l'hopital, but is struggling to solve it without using that method. They have attempted to multiply the numerator and denominator by (sqrt(3-x) + 1), but have not been successful in finding the answer.
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Homework Statement


lim x→2 ((sqrt(6-x)-2) / (sqrt(3-x) - 1)


Homework Equations


none


The Attempt at a Solution


My daughter's grade 12 mathematics class is not yet using l'hopital, yet this question has been posed as a practice question. I can solve it readily with l'hopital, however without it I'm stumped. When I attempt by multiplying numerator and denominator by (sqrt(3-x) +1 ) predictably I get 0 for my troubles.
 
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FAQ: Finding a limit without l'hopial

How do I find a limit without using l'Hopital's rule?

There are several methods for finding a limit without using l'Hopital's rule. One method is to use algebraic manipulation to simplify the expression and then substitute the limit value. Another method is to use the squeeze theorem to find a lower and upper bound for the expression and show that they both approach the same limit.

When should I use l'Hopital's rule to find a limit?

L'Hopital's rule should only be used when the limit is in an indeterminate form, such as 0/0 or ∞/∞. If the limit is not in an indeterminate form, then it is not necessary to use l'Hopital's rule.

Can I use l'Hopital's rule for all types of functions?

No, l'Hopital's rule can only be used for functions that are differentiable. This means that the function must be continuous and have a defined derivative at the point where the limit is being evaluated.

Is l'Hopital's rule the only method for finding limits?

No, l'Hopital's rule is just one method for finding limits. There are other methods, such as using algebraic manipulation, the squeeze theorem, and the limit definition of derivatives, that can be used to find limits.

Are there any limitations to using l'Hopital's rule?

Yes, there are some limitations to using l'Hopital's rule. For example, it can only be used for limits that are in an indeterminate form and it may not always give the correct answer. Additionally, it may not be applicable for certain types of functions, such as logarithmic and exponential functions.

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