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