Cal exam in 6 hours. its 3 am...
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Let L = lim[x→∞] (x/(x+1))^x
ln(L) = lim[x→∞] ln((x/(x+1))^x)
ln(L) = lim[x→∞] x ln(x/(x+1))
Let u = x/(x+1)
Then x = u/(1−u)
As x→∞, u = x/(x+1)→1
ln(L) = lim[u→1] u/(1−u) ln(u)
ln(L) = lim[u→1] u ln(u) / (1−u) = 0/0 -----> Use L'Hopital's Rule
ln(L) = lim[u→1] (u*1/u + 1*ln(u)) / −1
ln(L) = lim[u→1] (1 + ln(u)) / −1
ln(L) = (1 + ln(1)) / −1 = (1+0) / −1
ln(L) = −1
L = e^(−1)
L = lim[x→∞] (x/(x+1))^x = 1/e
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@Mike, when x gets very large, then (x/(x+1))^x gets close to 0.367879441 (1/e), not 0
x = 1 -----> (1/2)^1 = 0.5
x = 10 ----> (10/11)^10 = 0.385543289
x = 100 ----> (100/101)^100 = 0.369711212
x = 1000 ----> (1000/1001)^1000 = 0.368063304
x = 10000 ----> (10000/10001)^10000 = 0.367897834
x = 100000 ----> (100000/100001)^100000 = 0.367881281
x = 1000000 ----> (1000000/1000001)^1000000 = 0.367879625
http://www.google.ca/#hl=en&sclient=psy-…
ln(L) = lim[x→∞] ln((x/(x+1))^x)
ln(L) = lim[x→∞] x ln(x/(x+1))
Let u = x/(x+1)
Then x = u/(1−u)
As x→∞, u = x/(x+1)→1
ln(L) = lim[u→1] u/(1−u) ln(u)
ln(L) = lim[u→1] u ln(u) / (1−u) = 0/0 -----> Use L'Hopital's Rule
ln(L) = lim[u→1] (u*1/u + 1*ln(u)) / −1
ln(L) = lim[u→1] (1 + ln(u)) / −1
ln(L) = (1 + ln(1)) / −1 = (1+0) / −1
ln(L) = −1
L = e^(−1)
L = lim[x→∞] (x/(x+1))^x = 1/e
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@Mike, when x gets very large, then (x/(x+1))^x gets close to 0.367879441 (1/e), not 0
x = 1 -----> (1/2)^1 = 0.5
x = 10 ----> (10/11)^10 = 0.385543289
x = 100 ----> (100/101)^100 = 0.369711212
x = 1000 ----> (1000/1001)^1000 = 0.368063304
x = 10000 ----> (10000/10001)^10000 = 0.367897834
x = 100000 ----> (100000/100001)^100000 = 0.367881281
x = 1000000 ----> (1000000/1000001)^1000000 = 0.367879625
http://www.google.ca/#hl=en&sclient=psy-…
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Sorry the answer is zero, I didnt see the exponent part of the problem....
The best way to figure an infinite limit out is to plug in 1 for x, then go big and plug something like one million in. The equation gets closer and closer to zero
The best way to figure an infinite limit out is to plug in 1 for x, then go big and plug something like one million in. The equation gets closer and closer to zero