Evaluate the limit #lim_(x rarr 0^+) (1+1/x)^x #?
2 Answers
Jul 23, 2017
Explanation:
Jul 23, 2017
# lim_(x rarr 0^+) (1+1/x)^x = 1 #
Explanation:
We seek:
# L = lim_(x rarr 0^+) (1+1/x)^x #
As the log function is monotonic we can tak logs of both sides to get:
# ln L = ln {lim_(x rarr 0^+) (1+1/x)^x} #
# \ \ \ \ \ \ = lim_(x rarr 0^+) ln {(1+1/x)^x} #
# \ \ \ \ \ \ = lim_(x rarr 0^+) xln (1+1/x) #
# \ \ \ \ \ \ = lim_(x rarr 0^+) xln ((x+1)/x) #
# \ \ \ \ \ \ = lim_(x rarr 0^+) x{ln (x+1) - lnx} #
# \ \ \ \ \ \ = lim_(x rarr 0^+) {xln (x+1) - xlnx} #
# \ \ \ \ \ \ = lim_(x rarr 0^+) xln (x+1) - lim_(x rarr 0^+)xlnx #
# \ \ \ \ \ \ = 0 - 0 #
Thus:
# L = e^0 = 1 #