Lately, I've been very confused about the weird properties of limits. For example, I was very surprised to find out that $\lim_{n \to \infty} (3^n+4^n)^{\large \frac 1n}=4$ , because if you treat this as an equation, you can raise both sides to the $n$ power, subtract, and reach the wrong conclusion that $\lim_{n \to \infty} 3^n=0$ . I've asked this question before over here, and the answer was that $\infty-\infty$ is not well defined. I also found out here that you cannot raise both sides of a limit to a power unless the limit is strictly less than $1$ . However, there are also many examples where limits are treated as equations. For example, taking the logarithm of each side is standard procedure. Substitutions such as using $\lim_{x \to 0} \frac {\sin x}{x}=1$ work (although other substitutions sometimes don't work). So when can a limit be treated as an equation? Can you take for example the sine or tangent of each side like you can take the log? My guess is that you can treat it as an equation $at$ $least$ whenever $nothing$ is approaching $0$ or $\infty$ , but I'm not sure. Thanks.
P.S. Please keep the answers at a Calculus 1 level, and I have not learned the epsilon-delta definition of a limit.
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$\begingroup$The $n$ in the limit has no meaning outside of the limit. Therefore you cannot raise both sides to the $n$th power and then "bring in" the $n$ from the outside into the limit (to be combined with the $1/n$ exponent, as you seem to be doing). In an expression such as $$\lim_{n \to \infty} (3^n+4^n)^{1/n}$$ $n$ is a "dummy" variable. It simply tells you which variable in the inner expression we are taking the limit with respect to. The value of the limit is not a function of $n$, and therefore "raising both sides to the $n$th power" is not a meaningful operation.
It is, however, okay to square both sides (or cube, or raise to any fixed power) of the equation, or take the log, sin, tan, or any other function. An equation involving a limit is still an equation and you can always do any operation to both sides of an equation. However, you still have to be careful. Suppose you have an equation like $$\lim_{x \to 0} f(x) = L.$$ Now you take the log of both sides. You get $$\log \lim_{x \to 0} f(x) = \log L.$$ Notice I have not yet brought the $\log$ "into" the limit, which is usually the next step you want to take. In order for that step to be valid, you need to know that $\log$ is continuous at the values of $f(x)$ near $x=0$, and then you would get $$ \lim_{x \to 0} \log f(x) = \log L.$$
$\endgroup$ 7 $\begingroup$This is far from a complete answer, but I will try to address what I think might be the main problem.
If the function $f$ is continuous at the limit, then we have $$ \lim_{n\to\infty}f\left(\left(3^n+4^n\right)^{1/n}\right)=f(4)\tag{1} $$ That is we can pass $f$ across the limit. However, since $n$ varies as part of the limit (that is, it is bound to the limit as a dummy variable), we cannot use $f(x)=x^n$ to get $$ \lim_{n\to\infty}3^n+4^n=4^n\tag{2} $$ The $n$ in $\lim\limits_{n\to\infty}\left(3^n+4^n\right)^{1/n}$ is only meaningful inside the limit. If we write $\left(\lim\limits_{\color{#00A000}{n}\to\infty}\left(3^{\color{#00A000}{n}}+4^{\color{#00A000}{n}}\right)^{1/\color{#00A000}{n}}\right)^\color{#C00000}{n}$, the green $n$ tends to $\infty$ whereas the red $n$ does not. The green $n$ is a dummy variable and could just as easily be replaced by any other variable, and it has no fixed value. The red $n$ affects the value of the expression, and it needs a value for the expression to have a value.
I see that in this answer, the validity of $$ \lim_{n\to\infty}f_n(x_n)=\lim_{k\to\infty}f_k(\lim_{n\to\infty}x_n)\tag{3} $$ is considered. If $f_k\to f$ uniformly in a neighborhood of $\lim\limits_{n\to\infty}x_n$, and $f$ is continuous at $\lim\limits_{n\to\infty}x_n$, then $(3)$ is valid.
Since $f_k(x)=x^k$ converges uniformly to $0$ on $[-1+\epsilon,1-\epsilon]$ for $\epsilon>0$, if $\lim\limits_{n\to\infty}x_n\in(-1,1)$, then both sides of $(3)$ tend to $0$.
However, it should be understood that there is absolutely no connection between the index of the outer limit and the index of the inner limit on the right side of $(3)$. In fact for each $k$, $n\to\infty$.
If we let $f_k(x)=x^k$ and $x_n=(3^n+4^n)^{1/n}$, $(3)$ becomes $$ \lim_{n\to\infty}\left((3^n+4^n)^{1/n}\right)^n=\lim_{k\to\infty}\left(\lim_{n\to\infty}(3^n+4^n)^{1/n}\right)^k\tag{4} $$ However, since $\lim\limits_{n\to\infty}(3^n+4^n)^{1/n}=4$ and $f_k$ does not converge uniformly in a neighborhood of $4$, $(4)$ does not hold, and we cannot assume $k=n$ or think of the right side as $\lim\limits_{n\to\infty}3^n+4^n$, especially since that limit is not finite.
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