Question

How do you find lim√x2+x−x as x→∞?

Answer 1

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The initial form for the limit is

indeterminate ∞−∞

So, use the conjugate.

(√x2+x−x)=√x2+x−x1⋅√x2+x+x√x2+x+x

=x2+x−x2√x2+x+x

=x√x2+x+x

lim x→∞ x√x2+x+x has indeterminate form ∞ ∞, but we can factor and reduce.

We know that √x2=|x|, so for positive x (which is all we are concerned about for a limit as x increases without bound) we have

x√x2+x+x=x√