1 raised to Infinity becomes finite???

On limit studies, $1^\infty$ is not determined.

I think that happens because this limit depends on who approaches "1" faster: the basis or the exponent. It may result 0 if the exponent is much faster and the basis goes to 1 from above. It may result 0 if the exponent is faster and the basis approaches from below...

So...

a) It is not a number. In fact, $\infty$ is not a real number (maybe an extended one), so the only way to define $1^\infty$ is using limit process.

b) The error is not in the arguments, but in it's implications for calculus. We just can't define this limit properly.

c) You can find some exemples of limits giving indetermination $1^\infty$ and in fact may result 0, $\infty$, 1...

$\lim_{n \rightarrow \infty} 1^n = 1$ but if you are looking for $\lim_{n \rightarrow \infty} f(n)^{g(n)}$ where $\lim_{n \rightarrow \infty} f(n) = 1$ and $\lim_{n \rightarrow \infty} g(n) = \infty$, it depends on the specific $f$ and $g$.

First of all, $1^{\infty}$, in this case, is not a number. It is what mathematicians call an indeterminate form (because it can't be uniquely determined what it is).

It could be represented by something of the form $\lim_{x \to a} {f(x)^{g(x)}}$, where $f(x)$ and $g(x)$, are functions such that when $x \to {a}$ (for $a \in \mathbb{R}$ or $a = \pm \infty$), $f(x) \to 1$ and $g(x) \to \infty$.

One of the standard ways in which its value can be resolved (and in general, whether it is convergent or divergent) is to convert it into an indeterminate form of $\frac{0}{0}$ or $\frac{\infty}{\infty}$ by doing this neat trick:

$\lim_{x \to a} {f(x)^{g(x)}}$ = $\exp(\lim_{x \to a} {\frac{\ln{f(x)}}{1/g(x)}})$

or

$\lim_{x \to a} {f(x)^{g(x)}}$ = $\exp(\lim_{x \to a} {\frac{g(x)}{1/\ln{f(x)}}})$

These limits can then be standardly evaluated using l'Hôpital's rule. For example, in the case of $\lim_{x \to 0} {(1+x)^{1/x}}$:

$\lim_{x \to 0} {(1+x)^{1/x}}$ = $\exp(\lim_{x \to 0} {\frac{\ln{(1+x)}}{x}})$ = $\exp(\lim_{x \to 0} {\frac{1/(1+x)}{1}})$ (using l'Hôpital's rule) = $\exp(1)$ = $e$.

Of course, this all changes if by 1 you mean the constant function $f(x) = 1$. I think that in that case this indeterminate form evaluates to 1 every time.

To know whether 1 ^ ∞ = 1 or not, we need to understand the meaning of ∞ We all know that _ ∞ _ is not defined, but why is it so ?

Rather it should be defined (according to me). Now I tell you something, try it, do it : Ask yourselves a few questions -

• How many points are there on a line segment of length 2 cm ?
• How many points are there on a line segment of length 10 cm ?

Of course your answers will be the same, but do you think it should be the same i.e. there are same no. of points on line segment of 2 cm and that of 10 cm but then what's the point of difference between them ??

First of all, $\infty$ is not a real or a complex number, so you should not appeal to properties of those number systems to evaluate expressions with infinity. You can sometimes study expressions with infinity by interpreting it as a limit, but that is a human reinterpretation of the question.

Even if this question were interpreted as a limit, the most direct translation would be to: $\lim_{x \to \infty} 1^x$ And this limit does equal 1.

A less direct interpretation would be to interpret it as: $\lim_{a \to 1 \text{and} x \to \infty} a^x$ This expression is undefined unless there's a relationship between a and x.

There are some number systems (such as the surreal numbers) which do include arithmetic properties of infinity, but you will have to be clear about what number system you intend, and understand the properties of that number system when evaluating the expression. In the surreal number system, I believe $1^{\omega}$ evaluates to 1.

please explain a bit clear even i have the same doubt

pls!!

1^∞ is an indeterminate term because for limit we take left hand(lhl) as well right hand limit(rhl). For this case lhl is 0 as 1-h where h tends 0 is term smaller than 1 and on repeated multiplication will go on to 0 whereas rhl is ∞ because 1+h where h tends 0 is greater than 1 and on repeated multiplication goes to ∞....

1 raised to any power is 1, always.

We already discussed this a week and a half ago. It shouldn't be too far down: Page 3 or so. read that.

I think everything has $1^{\infty}$ as their identity.

Look at my example below.

If we have number $2$. Then we write like this $2 \times 1$. After that, we also write like this $2 \times 1 \times 1$, like this too $1 \times 2 \times 1 \times 1$ and so on or we simplify like this $2 \times 1^{\infty}$ and like this $1^{\infty} \times 2 \times 1^{\infty}$ and so on again.

isn't it?