You are watching: 1 to the power of infinity

It isn’t: $lim_n oinfty1^n=1$, specifically as friend suggest. However, if $f$ and $g$ are attributes such that $lim_n oinftyf(n)=1$ and $lim_n oinftyg(n)=infty$, the is not have to true that
$$lim_n oinftyf(n)^g(n)=1;. ag1$$
For example, $$lim_n oinftyleft(1+frac1n ight)^n=eapprox2.718281828459045;.$$
More generally,
$$lim_n oinftyleft(1+frac1n ight)^an=e^a;,$$
and as $a$ ranges over all real numbers, $e^a$ varieties over all hopeful real numbers. Finally,
$$lim_n oinftyleft(1+frac1n ight)^n^2=infty;,$$
and
$$lim_n oinftyleft(1+frac1n ight)^sqrt n=0;,$$
so a border of the type $(1)$ constantly has to be evaluate on its very own merits; the boundaries of $f$ and also $g$ don’t by themselves determine its value.
re-superstructure
point out
follow
answered Mar 3 "13 in ~ 20:29

Brian M. ScottBrian M. Scott
579k5151 yellow badges683683 silver- badges11211121 bronze badges
$endgroup$
include a comment |
10
$egingroup$
The limit of $1^infty$ exist:$$lim_n oinfty1^n$$ is not indeterminate. However$$lim_a o 1^+,n oinftya^n$$ is indeterminate..
re-publishing
point out
follow
edited Apr 15 "18 in ~ 4:43

KingLogic
1,33466 gold badges1010 silver badges2727 bronze title
answered Mar 3 "13 at 20:21
user35039user35039
$endgroup$
add a comment |
5
$egingroup$
There are plenty of reasons. Because that example, permit $1^infty=1$. Taking logarithm, you have $inftycdot 0=0$. Likewise for various other operations you will acquire some absurd.
re-superstructure
point out
follow
answer Mar 3 "13 in ~ 20:25

Boris NovikovBoris Novikov
16.9k11 yellow badge1818 silver badges3030 bronze title
$endgroup$
1
add a comment |
3
$egingroup$
The quick answer is that it is due to the fact that $x^y$ deserve to tend to any nonnegative limit together $x o1$ and $y oinfty$. For one example, consider the timeless limit $$lim_n oinftyBigl(1+frac xnBigr)^n=e^x.$$
re-publishing
point out
follow
edited Apr 13 "18 at 0:27

Paul Sinclair
34.8k22 yellow badges2222 silver- badges5757 bronze title
answer Mar 3 "13 in ~ 20:20
Harald Hanche-OlsenHarald Hanche-Olsen
30.4k22 gold badges5252 silver- badges7777 bronze title
$endgroup$
0
include a comment |
Not the price you're feather for? Browse other questions tagged limits infinity indices or asking your own question.
Upcoming occasions
Featured ~ above Meta
Linked
167
Why is $1^infty$ thought about to be an indeterminate form
4
Confusion end $a_k = (1+frac1k)^k = e$
related
3
Why is $fracsum_n=1^infty nsum_n=1^infty n$ indeterminate?
1
limit on two variables pull close infinity
2
evaluating limits at positive and negative infinity
2
show the limit of a duty doesn't exist
4
Evaluate border of the series: $lim_n oinfty left
1
The Power legislation of limits at infinity
warm Network inquiries much more hot questions
hunterriverpei.comematics
agency
stack Exchange Network
site architecture / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.11.10.40715
See more: One Difference Between Mitochondria And Chloroplasts Is, Difference Between Mitochondria And Chloroplasts
hunterriverpei.comematics ridge Exchange works best with JavaScript enabled

her privacy
By click “Accept all cookies”, friend agree stack Exchange deserve to store cookies on your maker and disclose info in accordance with our Cookie Policy.