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

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Answer 1 · 2014-08-30T06:14:12

By using logarithm and l'Hopital's Rule,
$lim_{x to infty}(1+a/x)^{bx}=e^{ab}$ .

By using the substitution $t=a/x$ or equivalently $x=a/t$ ,
$(1+a/x)^{bx}=(1+t)^{{ab}/t}$

By using logarithmic properties,
$=e^{ln[(1+t)^{{ab}/t}]}=e^{{ab}/t ln(1+t)}=e^{ab{ln(1+t)}/t}$

By l'Hopital's Rule,
$lim_{t to 0}{ln(1+t)}/{t}=lim_{t to 0}{1/{1+t}}/{1}=1$

Hence,
$lim_{x to infty}(1+a/x)^{bx}=e^{ab lim_{t to 0}{ln(1+t)}/{t}}=e^{ab}$

(Note: $t to 0$ as $x to infty$ )
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