let lim
Post on 18-Nov-2015
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Let lim
TRANSCRIPT
Let lim_{x?8} (1 + 1/x)^x = L [we are not requiring |L| < 8]. L = lim_{x?8} (1 + 1/x)^x, ln(L) = ln(lim_{x?8} (1 + 1/x)^x), ln(L) = lim_{x?8} ln((1 + 1/x)^x), ln(L) = lim_{x?8} xln(1 + 1/x), ln(L) = lim_{x?8} ln(1 + 1/x)/(1/x) [=0/0, so we apply l'Hopital's rule], ln(L) = lim_{x?8} ((-1/(x^2))/(1 + 1/x)) / (-1/(x)^2), ln(L) = lim_{x?8} 1/(1 + 1/x), ln(L) = lim_{x?8} x/(x + 1), ln(L) = 1, e^ln(L) = e^1, L = e, lim_{x?8} (1 + 1/x)^x = e.
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