I saw in the book Calculus 6th Edition by J. Stewart this law of limit:
$$ \text{6. }\lim\limits_{x\to a} [f(x)]^n = \left[ \lim_{x\to a} f(x) \right]^n, \text{ $n$ is a positive integer, (p. 101, Sec 2.3).} $$ And in page 133 of the same book: 'It can be proved that the Limit Laws listed in Section 2.3 (with the exception of Laws 9 and 10) are also valid if "$x \to a$" is replaced by "$x\to \infty$" or "$x\to-\infty$".'
The Law 6 above is, therefore, valid for limit at infinity:
$$ \lim_{x\to \infty} [f(x)]^n = \left[\lim_{x\to \infty} f(x)\right]^n, \text{ $n$ is a positive integer.} $$
However there is something I do not know of and have not found in the same book. That thing is:
$$ \lim_{x\to \infty} [f(x)]^t = \left[\lim_{x\to \infty} f(x) \right]^t, \text{ $t$ is real and positive} \tag{$*$} $$
Is this really true? I read many proofs in which people imply using this law even in the same book. I'm undergraduate student so maybe I haven't taught this yet but I really want to know because of the usefulness of the law $(*)$ if it should be true. I'm very new to this forum and I'm not a native English speaker so apologize my grammar and vocabulary.
If $f(x)$ is positive (or eventually positive as $x \to \infty$), then this is true, as we are invoking the continuity of the function $x^t$ defined for positive $x$. If $f$ takes on negative values or the limit is negative, you can't apply this anymore since raising a negative number to a real exponent is not defined (in real-variable calculus).