Why is it that when we are finding the limit as x tends to infinity of a function, we disregard all variables on the numerator and denominator, except for the ones with the highest degree?

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2 Answers

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Heuristically, it is because terms of higher order degree get bigger way, way faster than terms of lower order degree.

Mathematically, try dividing the numerator and denominator by npn^p where pp is the largest degree present in the expression. This is just multiplying by 1 and should make what happens in the limit clear.

Because, let for example the numerator be xm+xm−1+…+1x^m + x^{m-1}+…+1 you can write it as xm(1+1x+…+1xm)x^m (1+\frac 1 {x} + … + \frac 1 {x^m}) . As x tends to ∞\infty the numerator tends to xmx^m for the theorem of limit product.