Csdrhrt

Evaluate $$lim_{n to infty} frac{(1+sqrt 2)^n+(1-sqrt 2)^n}{(1+sqrt 2)^n-(1-sqrt 2)^n}.$$

I tried to expand using Newton's Binomial Theorem, but it didn't work.

$lim_{n to infty} frac{(1+sqrt 2)^n+(1-sqrt 2)^n}{(1+sqrt 2)^n-(1-sqrt 2)^n}$
$= lim_{nto infty }frac{1+a^n}{1-a^n} $ such that
$a=frac{(1-2^{frac{1}{2}})}{(1+2^{frac{1}{2}})} $ , $|a|<1$. So
lim is equal to $frac{1-0}{1+0}=1$

Roughly, $|1-sqrt 2| lt 1$, so a high power of it will go to $0$. $1+sqrt 2 gt 1$, so a high power of it will be large and positive. We can ignore the two small terms and be left with the fixed ratio $1$. Depending on what theorems you have proved about limits that may be enough.

We have that

• $|1-sqrt 2|<1 implies (1-sqrt 2)^n to 0$

therefore

$$frac{(1+sqrt 2)^n+(1-sqrt 2)^n}{(1+sqrt 2)^n-(1-sqrt 2)^n}sim frac{(1+sqrt 2)^n}{(1+sqrt 2)^n}=1$$

or more rigoursly

$$frac{(1+sqrt 2)^n+(1-sqrt 2)^n}{(1+sqrt 2)^n-(1-sqrt 2)^n}= frac{(1+sqrt 2)^n}{(1+sqrt 2)^n}frac{1+frac{(1-sqrt 2)^n}{(1+sqrt 2)^n}}{1-frac{(1-sqrt 2)^n}{(1+sqrt 2)^n}}to frac{1+0}{1-0}$$