Thus, it is possible to evaluate a limit when the denominator approaches 0, because we are not actually dividing by 0, but by something exceedingly close to 0.Īnyway, here are the various scenarios for the limit as x→∞ for rational functions (that is what you call a polynomial divided by a polynomial).
This distinction is important because having a denominator approaching 0 is not the same thing as having a denominator actually at 0. Is this when some sort of algebraic simplification is required so as to determine the limit as the function approaches infinity? Or is there a massive flaw in my reasoning? Thank you :)įirst, remember that with limits we are not evaluating what the function is at the limiting value, but rather what the function approaches as we get infinitesimally close to the limiting value. Going further into that, if you employ the neat little trick of dividing all the terms in the function by the highest degree power of x, the denominator would be seen to approach 0 as we get to infinity and we know that we cannot have a denominator value of 0. Is it safe to assume that the limit as x -> ∞ of any standard polynmial in the case of a polynomial fraction (so to speak) where the highest degree power of the numerator is equal to that of the denominator it is the ratio of the coefficient of the highest degree powers in both the numerator & denominator? And the case where the numerator's highest degree is less than the denominator's the limit is 0 (i.e the funtion is approaching 0 as x -> ∞) because the denominator overpowers the numerator? But then if the highest degree power in the denominator is less than the highest degree power in the numerator, the denominator will obviously be overpowered by the numerator.
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