How do you find the limit of ${(tan 4 x)}^{x}$ $(Tan4x)^x$ as x approaches 0?

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Answer 1 · 2017-09-19T00:14:34

$lim_(x rarr 0) (tan4x)^x =1$

We seek:

$L = lim_(x rarr 0) (tan4x)^x$

The logarithmic function is monotonically increasing so we have:

$ln L = ln{lim_(x rarr 0) (tan4x)^x }$
$\ \ \ \ \ \ = lim_(x rarr 0) ln{(tan4x)^x }$
$\ \ \ \ \ \ = lim_(x rarr 0) xln(tan4x)$

Note that both $tanx rarr 0$ and $x rarr 0$ uniformly as $x rarr 0$ , and so:

$ln L = 0 xx 0$
$\ \ \ \ \ \ = 0$

And so:

$L = e^ 0 =1$

How do you find the limit of (Tan4x)^x(tan4x)x(Tan4x)^x as x approaches 0?