09.30 - Section 1.6 Limits involving infinity Key idea Even...

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Unformatted text preview: Section 1.6: Limits involving infinity Key idea: Even though infinity isn’t 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 δ > 0 such that) (if 0 < | x – a | < δ ) then f ( x ) > M . In the specific case f ( x ) = 1/ x 2 , what δ can we pick as a function of M ? ..?.. δ = 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 δ > 0 such that) (if 0 < | x – a | < δ ) then f ( x ) < M . Sometimes we write ∞ as + ∞ to make the distinction with – ∞ clear. Trigonometric example: lim x → π /2– tan( x ) = + ∞ , lim x → π /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 ) = π /2, lim x → – ∞ arctan( x ) = – π /2....
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09.30 - Section 1.6 Limits involving infinity Key idea Even...

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