I know infinity is not a number so is this question undefined or something Can anyone explain it to me please
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limits - 1 to the power of infinity, why is it indeterminate? - Mathematics Stack Exchange
Is infinity +1 greater than infinity?
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It is worth noting that since 
is not a number, the expression 
is not meaningful. That is, you cannot evaluate this expression. It is not a number. You can think of it in much the same way as 
--- an expression without any interpretation.
The only means that we have to talk about expressions involving 
is through the concept of limits. The shorthand
is sometimes used in a careless way by students, but the middle expression in this string of equations isn't strictly meaningful. It is only a device used to remind you that we are looking at what happens as we divide 1 by increasingly large numbers.
So, I suppose that in some sense, you can say that 
, as the left-hand side here isn't any number at all.
Your argument makes use of stochastics. If you want to do things the mathematically exact way, you would first need to define a probability measure on 
. Assuming that you choose a continuous distribution, then yes, the Probability that you hit 
is 
- but there are infinitely many numbers, so there is no contradiction.
(Keep in mind that 
is not well defined, and unless in the case of 
there is no heuristical way to define it)
Check out Measure Theory: https://en.wikipedia.org/wiki/Measure_%28mathematics%29
What 
is, or is not, is merely a matter of definition. Normally, one would only define 
for some specific class of pairs of 
- say 
- positive integer, 
- real number.
When extending the definition of exponentiation to more general pairs, the key thing people keep in mind is that various nice properties are preserved. For instance, for 
- positive integer, you want to put 
so that the rule 
is preserved.
It may make sense in some context to speak of infinities in the context of limits, but this is usually more a rule of thumb than rigorous mathematics. This may be seen as extending the rule that 
is continuous (i.e. if 
and 
, then 
) to allow for 
. For instance, you may risk saying that:
If you agree to use rules of this kind, you might be tempted to also say:
but this would lead you astray, since in reality:
Thus, it is safer to leave 
undefined.
A more thorough discussion can be found on Wikipedia.
When your teacher talks about 
or 
or 
he/she's not talking about numbers, but about functions, more precisely about limits of functions.
It's just a convenient expression, but it should not be confused with computations on simple numbers (which 
isn't, by the way).
When 
is referred to, it is to mean the following situation: there are two functions 
and 
defined in a neighborhood of 
, with the properties
(or 
)
(of course, 
can also be 
or 
).
Saying that 
is an indeterminate form is just a mnemonic way to say that you cannot compute
just by saying “the base goes to 
, so the limit is 
because 
”. Indeed this can be grossly wrong as the fundamental example
shows.
Why is that? It's easy if you always write 
as 
and compute the limit of 
, then applying the properties of the exponential function.
In the case above we'd have
(or 
)
so the limit
is in the other 
indeterminate form (that you should know). Why is it “indeterminate”? Because we have many instances of that form where the limit is not predictable by simply doing a (nonsense) multiplication:
\begin{gather} \lim_{x\to 0+}x\cdot\frac{1}{x}=1\\ \lim_{x\to 0+}x^2\cdot\frac{1}{x}=0\\ \lim_{x\to 0+}x\cdot\frac{1}{x^2}=\infty \end{gather}
It isn’t: $\lim_{n\to\infty}1^n=1$, exactly as you suggest. However, if $f$ and $g$ are functions such that $\lim_{n\to\infty}f(n)=1$ and $\lim_{n\to\infty}g(n)=\infty$, it is not necessarily true that
$$\lim_{n\to\infty}f(n)^{g(n)}=1\;.\tag{1}$$
For example, $$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e\approx2.718281828459045\;.$$
More generally,
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{an}=e^a\;,$$
and as $a$ ranges over all real numbers, $e^a$ ranges over all positive real numbers. Finally,
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}=\infty\;,$$
and
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{\sqrt n}=0\;,$$
so a limit of the form $(1)$ always has to be evaluated on its own merits; the limits of $f$ and $g$ don’t by themselves determine its value.
The limit of $1^{\infty}$ exist:$$\lim_{n\to\infty}1^n$$ is not indeterminate. However$$\lim_{a\to 1^+,n\to\infty}a^n$$ is indeterminate..