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Unformatted text preview: Section 1.6: Limits involving infinity Key idea: Even though infinity isnt a real number, it can be useful as a way of indicating how functions behave. E.g., for f ( x )=1/ x 2 , according to our earlier definition, lim x 0 1/ x 2 is undefined, but we find it useful to write lim x 1/ x 2 = + . More generally, we write lim x a f ( x ) = to mean that (for all M ) (there exists > 0 such that) (if 0 < | x a | < ) then f ( x ) > M . In the specific case f ( x ) = 1/ x 2 , what can we pick as a function of M ? ..?.. = 1/sqrt( M ). Note that this holds for f ( x )=1/ x 2 and f ( x )=1/ x 4 (with a = 0) but not for f ( x )=1/ x or f ( x )=1/ x 3 [have someone say why]. We can however write lim x 0+ 1/ x = or lim x 0 1/ x = . We write lim x a f ( x ) = to mean that (for all M ) (there exists > 0 such that) (if 0 < | x a | < ) then f ( x ) < M . Sometimes we write as + to make the distinction with clear. Trigonometric example: lim x /2 tan( x ) = + , lim x /2+ tan( x ) = . These sorts of limiting assertions correspond to vertical asymptotes of the graph. Similarly for horizontal asymptotes: E.g., for the arctan function, we write lim x arctan( x ) = /2, lim x arctan( x ) = /2....
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