I'm looking to evaluate
limx→+∞√x2+4x+1−x
The answer in the book is 2. How do I simply evaluate this problem?
I usually solve limits such as this with the short cut method, i.e (Numerator degree < Denominator degree) = 0 ; (Numerator degree = Denominator degree )= take ratio of leading coefficients; (Degree numerator > degree denominator )= take leading terms and use algebra to simplify and then plug in −∞ or +∞
Please keep in mind that I do not know L'Hopital's rule.
Answer
limx→∞√x2+4x+1−x=limx→∞(√x2+4x+1−x)⋅√x2+4x+1+x√x2+4x+1+x=limx→∞(x2+4x+1)−x2√x2+4x+1+x=limx→∞4+1x√1+4x+1x2+1=41+1
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