What is $lim_(xrarroo) ( 1 + ( r/x ) ) ^ x$ , where r is any real number?

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Answer 1 · 2015-10-27T02:30:45

$lim_(xrarroo)(1+r/x)^x = e^r$

$(1+r/x)^x = ((1+r/x)^(x/r))^r$

Let $u = rx$ .so this becomes $((1+1/u)^u)^r$

Note that as $xrarroo$ , we also have $u rarroo$

So,

$lim_(xrarroo(1+r/x)^x = lim_(urarroo)((1+1/u)^u)^r$

$= (lim_(urarroo)(1+1/u)^u)^r$

$= e^r$

What is lim_(xrarroo) ( 1 + ( r/x ) ) ^ xlim_(xrarroo) ( 1 + ( r/x ) ) ^ x, where r is any real number?