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find the limit:

$$\lim_{ x \to0^+ }(\ln x)^{\cot x}$$

my try :

$$\lim_{x\rightarrow x_0}{f(x)^{g(x)}}=\left( \lim_{x\rightarrow x_0}{f(x)}\right)^{\left( \lim_{x\rightarrow x_0}{g(x)}\right)}$$

$$\lim_{ x \to0^+ }(\ln x)^{\cot x}=(\lim_{ x \to0^+ }(\ln x))^{\lim_{ x \to0^+ }(\cot x)}$$

now ?

  • 1
    The identity you are trying to use is not always valid! – 2017-02-16
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    The function is not well defined in the neighborhood of $0$. – 2017-02-16

2 Answers 2

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The limit doesn't exist. Notice that when $x\in(0,1)$, then $\ln(x)$ is negative.

Likewise, when $x\in(0,1)$, $\cot(x)$ is not always an integer.

Thus, you are getting negative numbers to non-whole exponents, which is undefined.

When $\cot(x)$ is an integer, depending on if it is even or odd, the limit goes to positive or negative infinity, so the limit doesn't exist for $\cot(x)\in\mathbb N$ either.

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Note that the right neighbourhood of the point is not in the domain of the function, so the limit does not exist.