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What is the limit of(1+(1/x))^x as x approaches infinity?

1 Answer
Aug 6, 2016

Answer:

Make the limit of (1+(1/x))^x as x approaches infinity equal to any variable e.g. y, k. and take the natural logarithm of both sides.

Explanation:

#y=lim_(x-oo)(1+(1/x))^x#
#ln y =lim_(x-oo)ln (1+(1/x))^x#
#ln y =lim_(x-oo)x ln (1+(1/x))#
#ln y =lim_(x-oo) ln (1+(1/x))/x^-1#
if x is substituted directly, the value will be undefined, so
l'hopital's rule is applied.
l'hopital's rule says that if #lim_(x-a) f(x) =0= lim_(x-a) g(x)#,
then #lim_(x-a)(f(x)/g(x)) = lim_(x-a) ((f'(x))/(g'(x)))#
#ln y =lim_(x-oo)((1/(1+(1/x)))(0-1x^-2))/(-1x^-2)#
#ln y =lim_(x-oo)(1/(1+(1/x)))#
substitute for x
#ln y = (1/(1+0))#
#ln y = 1#
introduce exponential #e#
#e^ln y = e^1#
#y = e#
#y = e = lim_(x-oo)(1+(1/x))^x#
#lim_(x-oo)(1+(1/x))^x = e#