Sunday, December 9, 2018

calculus - Limits: How to evaluate limlimitsxrightarrowinftysqrt[n]xn+an1xn1+cdots+a0x



This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates.






What methods can be used to evaluate the limit limxnxn+an1xn1++a0x.



In other words, if I am given a polynomial P(x)=xn+an1xn1++a1x+a0, how would I find limxP(x)1/nx.




For example, how would I evaluate limits such as limxx2+x+1x

or limx5x5+x3+99x+101x.


Answer



Your limit can be rewritten as
limx(n1+an1x++a0xn11x)


Or equivalently,
limy0(n1+an1y++a0yn1y)

This, by the definition of derivative, is the derivative of the function f(y)=n1+an1y++a0yn at y=0, which evaluates via the chain rule to an1n.


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