MATH 1A Lecture Notes - Lecture 10: Asymptote
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A: this is answered by f (x) lim x and lim x f (x) Here, f(x) gets arbitrarily large for sufficiently large x in either direction, so f (x) lim x lim x f (x) So the limit does not exist, but the fact that it increases without bound (i. e. goes to infinity ) shows us in which way it fails to exist. If we simply plug in , f (x) lim x ends up looking like. , so we need to rewrite this: f ( x) lim x x ( 2)(2+ 1 x2 ) (x2)(1+ 1 x2) lim x . For all finite x > 0, this becomes lim x (2+ 1 x2 ) (1+ 1 x2) We can see here that as x , the numerator tends to 2 and the denominator tends to 1 (since tends to 0 in the numerator and denominator). Meanwhile, since f is even, f(x) = f(-x).
