MIT18_014F10_pset10sols

MIT18_014F10_pset10s - 18.014 Problem Set 10 Solutions Total 12 points Problem 1 Evaluate log a be x a lim b lim log x log(1 − x x →∞ √ a

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Unformatted text preview: 18.014 Problem Set 10 Solutions Total: 12 points Problem 1: Evaluate log( a + be x ) a. lim b. lim log( x ) log(1 − x ). x →∞ √ a + bx 2 x → 1 − Solution (4 points) For part (a), we first evaluate the limits lim x →∞ log( a + be x ) and x lim x →∞ √ a + x bx 2 . The second limit can be computed as follows: x x 2 1 1 lim = lim = lim = . √ a + bx 2 a + bx 2 t + at 2 + b √ b x →∞ x →∞ → For the first limit, we use L’Hopital’s rule to get log( a + be x ) be x / ( a + be x ) b b lim = lim = lim = lim = 1 . x →∞ x x →∞ 1 x →∞ ae − x + b t → + ae t + b Now, we know lim f ( x ) g ( x ) = lim f ( x ) lim g ( x ) x →∞ x →∞ x →∞ whenever the two limits on the right hand side exist. Putting it all together, we have log( a + be x ) 1 lim = . x →∞ √ a + bx 2 √ b For part (b), note lim x → 1 − log( 1 x ) = ∞ and lim x → 1 − log(1 − x ) = ∞ . Using L’Hopital’s rule, we see log(1 − x ) − 1 / (1 − x ) x log(...
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This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

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MIT18_014F10_pset10s - 18.014 Problem Set 10 Solutions Total 12 points Problem 1 Evaluate log a be x a lim b lim log x log(1 − x x →∞ √ a

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