Monday, 1 May 2017

calculus - limntoinfty1n=1?




Is it really true that
lim



?



We were taught throughout our entire math courses that [1^{\infty}] is an undetermined form....Am I missing something here?


Answer




Is it really true that

\lim_{\color{red}{x}\to\infty} 1^n = 1




You probably mean:
\lim_{\color{blue}{n}\to\infty} 1^n = 1
and yes, this is true because 1^n = 1 for all n.



The expression "1^{+\infty}" is indeterminate and the limit above doesn't contradict that.



Perhaps you know the following well-known limit too:

\lim_{n\to\infty}\left(\color{blue}{1+\frac{1}{n}}\right)^\color{red}{n} = e \ne1where you also have \color{blue}{1+\frac{1}{n}}\to 1 and \color{red}{n}\to+\infty.



Combining both limits shows that you can have sequences c_n = \left(a_n\right)^{b_n} where a_n \to 1 and b_n\to+\infty but with different limits for c_n; which is why we call "1^{+\infty}" indeterminate.


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