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What methods can be used to evaluate the limit limx→∞n√xn+an−1xn−1+⋯+a0−x.
In other words, if I am given a polynomial P(x)=xn+an−1xn−1+⋯+a1x+a0, how would I find limx→∞P(x)1/n−x.
For example, how would I evaluate limits such as limx→∞√x2+x+1−x
or limx→∞5√x5+x3+99x+101−x.
Answer
Your limit can be rewritten as
limx→∞(n√1+an−1x+⋯+a0xn−11x)
Or equivalently,
limy→0(n√1+an−1y+⋯+a0yn−1y)
This, by the definition of derivative, is the derivative of the function f(y)=n√1+an−1y+⋯+a0yn at y=0, which evaluates via the chain rule to an−1n.
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