(Non)Existence of limits

When we say that a limit of a function does not exist in $\mathbb{R}$ (or some metric space) does it make sense to say that it might exist somewhere else? [I am trying to think along lines of existence of imaginary roots]

If yes. Then give examples especially regarding $\mathbb{R}$.

Solution 1:

Another example is $$\lim_{x\to 0}\frac1{x^2}$$ which does not exist in $\mathbb R$, but does exist in the extended reals $[-\infty,\infty]$.

It also exists in the extended complex numbers (Riemann sphere, roughly $\hat{\mathbb C}=\mathbb C\cup\{\infty\}$).

Note that $$\lim_{x\to 0}\frac1{x}$$ does not exist in the reals or the extended reals, but does exist in the Riemann sphere. That's because there is only one infinity in the Riemann sphere, but not so in the extended reals.

Solution 2:

Yes for example a sequence of rational numbers might converge to $\sqrt{2}$ which is not rational. . So the sequence does not converge in $\Bbb Q$ but does converge in $\Bbb R$.

If your sequence is in $\Bbb R$ though, then since $\Bbb R$ is complete, if the sequence is Cauchy then it does converge somewhere. So if it doesn't converge in $\Bbb R$ it won't converge anywhere bigger because it won't be Cauchy.

Solution 3:

For example the sequence $a_n = \bigg(1+\frac{1}{n}\bigg)^n$ has all $a_n \in \mathbb{Q}$ but $$\lim_{n\rightarrow \infty} \bigg(1+\frac{1}{n}\bigg)^n =e \notin \mathbb{Q} $$

If however, $a_n \in \mathbb{R}$ then $(\lim_{n\rightarrow \infty} a_n) \in \mathbb{R}$ (if it exists) since $\mathbb{R}$is complete.

Solution 4:

I am not sure if this might be helpful, but I bumped into the following example while answering this question Squeeze theorem on expression with minus. There, help was required for the following limit $$\lim_{x \rightarrow \infty }(-6e^{11x} + 9\sin(x) + 3e^{8x} )^{8/4x}\ .$$

The function $f(x)=(-6e^{11x} + 9\sin(x) + 3e^{8x} )^{8/4x}$ is not defined for $x$ real and large, because $h(x)=-6 e^{11x}+9\sin(x)+3 e^{8x}$ becomes badly negative there, so the limit is somehow ill-defined in $\mathbb{R}$. However, one might define a function $g:\mathbb{R}\to\mathbb{C}$ $$ g(x)=\exp\left(\frac{8}{4x}\mathrm{Log}(-6e^{11x} + 9\sin(x) + 3e^{8x} )\right)\ , $$ with $\mathrm{Log}$ the principal branch of the complex logarithm https://en.wikipedia.org/wiki/Complex_logarithm. The function $g(x)$ is 'morally' equivalent to $f(x)$ [and it coincides with $f(x)$ for all $x$ such that $h(x)>0$], but is perfectly well defined for all $x\in\mathbb{R}$. So, we might ask: what is $\lim_{x\to\infty}g(x)$ [the limit of a complex-valued function of a real argument]?

For $z=x+\mathrm{i}y$, using $\mathrm{Log}(z)=\ln |z|+\mathrm{i}\ \mathrm{atan2}(y,x)$, where $\mathrm{atan2}$ is defined in https://en.wikipedia.org/wiki/Atan2, one obtains $$ g(x)=\exp\left[\frac{8}{4x}\left(\ln(6e^{11x} - 9\sin(x) - 3e^{8x} )+\mathrm{i}\pi\right)\right]\ , $$ so that the imaginary part vanishes in the limit $x\to\infty$ and one obtains $$ \lim_{x\to\infty} g(x)=e^{22}\ , $$ the result given by Mathematica. Note, however, that Mathematica gives the same result for the limit $\lim_{x\to\infty}f(x)$, even though (as I remarked above) this is not defined for $x\in\mathbb{R}$. In summary, in this example we could give a meaning to an ill-defined limit of a real function by moving 'somewhere else' (i.e. allowing the function to take imaginary values for real arguments). Not sure if this was what the OP wanted to know, honestly.