I'm trying to solve evaluate this limit
limx→∞√x−1−√x−2√x−2−√x−3.
I've tried to rationalize the denominator but this is what I've got
limx→∞(√x−1−√x−2)(√x−2+√x−3)
and I don't know how to remove these indeterminate forms (∞−∞).
EDIT: without l'Hospital's rule (if possible).
Answer
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As x→∞ we can assume x>0 , so:
√x−1−√x−2√x−2−√x−3=√x−2+√x−3√x−1+√x−2=√1−2x+√1−3x√1−1x+√1−2x→x→∞1
Further hint: the first step was multiplying by conjugate of both the numerator and the denominator.
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