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Workshop limits - a a1 = 1 a2 = 2 a3 = 2.25 a4 = 2.3704 a5...

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a) a 1 = 1 a 2 = 2 a 3 = 2.25 a 4 = 2.3704 a 5 = 2.4414 a 6 = 2.4883 a 7 = 2.5216 a 8 = 2.5465 a 9 = 2.56571 a 10 = 2.58117 b 1 = 2 b 2 = 1.5625 b 3 = 1.3717 b 4 = 1.2744 b 5 = 1.2167 b 6 = 1.1787 b 7 = 1.1519 b 8 = 1.1321 b 9 = 1.1168 b 10 = 1.1046 c 1 = 2 c 2 = 2.9142 c 3 = 3.9245 c 4 = 5.0625 c 5 = 6.3483 c 6 = 7.7997 c 7 = 9.4334 c 8 = 11.267 c 9 = 13.318 c 10 = 15.607 From the following patterns, a n appears to approach e, b n appears to approach 1, and c n appears to approach infinity. b) Rewrite the first limit a n = (1+ 1/n) n in terms of x to find the limit as x tends to infinity. Therefore, the limit of f(x) as x approaches infinity, is equal to the limit as x approaches infinity of (1+1/x) x . To find the solution of the limit, let y = 1/x. Therefore, the limit as f(x) approaches infinity will equal the limit as y approaches 0 of (1+y) (1/y) .
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The natural log of both sides should be taken to find the limit of the integral. The 1/y as an exponent in the ln(1+y) (1/y) , can be written as a coefficient using the logarithm properties. Therefore, the limit as x approaches infinity of the natural log of f(x) = the limit as y approaches 0 of ln(1 +y )/y.
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