What is the limit of ${(1 + 2 x)}^{\frac{1}{x}}$ $(1+2x)^(1/x)$ as x approaches infinity?

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Answer 1 · 2017-02-16T09:31:45

$lim_(x->oo) (1+2x)^(1/x) = 1$

Write the function as:

$(1+2x)^(1/x) = (e^(ln(1+2x)))^(1/x) = e^(ln(1+2x)/x)$

Now evaluate:

$lim_(x->oo) ln(1+2x)/x$

This limit is in the indeterminate form $oo/oo$ so we can solve it using l'Hospital's rule:

$lim_(x->oo) ln(1+2x)/x = lim_(x->oo) (d/dx ln(1+2x))/(d/dx x) = lim_(x->oo) 2/(1+2x) = 0$

As $e^x$ is a continuous function we have then:

$lim_(x->oo) (1+2x)^(1/x) = lim_(x->oo) e^(ln(1+2x)/x) = e^((lim_(x->oo) ln(1+2x)/x)) = e^0 = 1$

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