How would you find this limit ?
$\lim_{n \to \infty}(1+\ln(1-\frac{1}{2n}))^n$
Thank you.
How would you find this limit ?
$\lim_{n \to \infty}(1+\ln(1-\frac{1}{2n}))^n$
Thank you.
We have $\lim\limits_{n \to \infty}(1+\ln(1-\frac{1}{2n}))^n = \lim\limits_{n \to \infty}\left((1+\ln(1-\frac{1}{2n}))^{\frac1{\ln(1-\frac{1}{2n})}}\right)^{n\ln(1-\frac{1}{2n})}= e^{\lim\limits_{n\to\infty} n\ln(1-\frac{1}{2n}) }$ provided the limit in the exponent exists. (Since $\lim\limits_{n \to \infty}\left((1+\ln(1-\frac{1}{2n}))^{\frac1{\ln(1-\frac{1}{2n})}}\right) = \lim\limits_{x\to 0} (1+x)^{\frac1x}=e$.)
So it only remains to compute the limit from the exponent. We get $\lim\limits_{n\to\infty} n\ln(1-\frac{1}{2n}) = -\frac12 \lim\limits_{n\to\infty} \frac{\ln(1-\frac{1}{2n})}{-\frac1{2n}}=-\frac12$ using $\lim\limits_{x\to 0}\frac{\ln(1+x)}x=1$.
So the original limit is $e^{-1/2}$.
A similar basic idea can be used in many computations of this type -- to rewrite this as limits containing expression $\left(1+\frac1{f(n)}\right)^{f(n)}$ where $f(n)\to\infty$ or $(1+g(n))^{\frac1{g(n)}}$ where $g(n)\to0$ and the compute the new limit in the exponent.
You can also use wolframalpha and click there on "show steps":
If there is any limit of the form $\lim_{x\rightarrow a}\left ( 1+f\left ( x \right ) \right )^{\frac{1}{g\left ( x \right )}}$such that$\lim_{x\rightarrow a}f\left ( x \right )=\lim_{x\rightarrow a}g\left ( x \right )=0$ It is given by $e^{\lim_{x\rightarrow 0}{\frac{f\left ( x \right )}{g\left ( x \right )}}}$
Take the limit of the logarithm:
$ \eqalign{ \lim_{n\rightarrow\infty } \ln [ 1+\ln (1-{1\over 2n})]^n &=\lim_{n\rightarrow\infty }{\ln [ 1+\ln (1-{1\over 2n})]\over 1/n}\cr &=\lim_{n\rightarrow\infty }{{1\over [ 1+\ln (1-{1\over 2n})]}\cdot{1\over 1-{1\over2n}}\cdot {1\over2n^2}\over -1/n^2}\cr &=\lim_{n\rightarrow\infty }{{1\over [ 1+\ln (1-{1\over 2n})]}\cdot{1\over 1-{1\over 2n}}\cdot {-1\over2 } }\cr &=-1/2. } $ So the original limit is $e^{-1/2}$.