MATH 104 Lecture Notes - Squeeze Theorem, Exponential Growth, Quadratic Function
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MATH 104 Full Course Notes
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Di erent functions have di erent growth rates, which are important for eval- uating limits at in nity. The function x2 grows faster than x: we have and lim xto x2 x. = lim x x = lim x x x2 = 0 and similarly, if n > m, then xn grows faster than xm. The growth rate of a polynomial comes from the growth rate of the top-degree term, and in general any sum grows as fast as its fastest-growing term. Let f (x) and g(x) be polynomials, with f (x) = adxd + . + a0 and g(x) = bexe + . + b0; by convention, we assume the top-degree terms ad and be are nonzero. We write their fraction as f (x) g(x) adxd + . + b0x e where we obtain the second equality by dividing each term by xe. If d = e, then we get lim x f (x) g(x)
