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Calculus/tough limit with radical
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Question
QUESTION: I am generally pretty good with evaluating and simplyfying expressions, but this one is a stumper:
Limit as x approaches 4
(sqrt(8-x)-2)/(sqrt(13-x)-3)
Any suggestions?
Thanks!
ANSWER: Remember that (a-b)(a+b) is a²-b²?
Note that the bottom of the fraction is a-b, so multiply by (a+b)/(a+b).
That is, multiply the top and bottom by √(13-x) + 3.
The bottom woud then be √(13-x)² - 3² = 13 - x - 9 = 4 - x.
---------- FOLLOW-UP ----------
QUESTION: The solution you gave is incorrect, as it divides by zero. It is the limit as x approaches 4. 4-x is zero. The reciprocal of either the numerator or the denominator causes the denominator to be zero.
Any other suggestions?
There is another approach to try.
We have (√(8-x) - 2)/(√13-x) - 3) as x approaches 4.
Calculus can be applied. Since f/g is 0/0, find f'/g', and that will be the value of the limit.
The problem gives us f(x) = √(x-2) - 2 and g(x) = √(13-x) - 3.
It can be seen that f'(x) = 1/(2√(x-2)) and g'(x) = -1/(2√(13-x)).
Both f'(4) and g'(4) can be evaluated.
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