I have been asked to find the following limit
$f(x)=(x^{2n}-1)/(x^{2n}+1)$ as n tends to infinity. The answers given are in MCQ type with more than one answer correct. I'll just write the correct answers.
a)$f(x)=1$,for$|x|>1$ and b) $f(x)=-1$, for $|x|<1$. I came up with first answer but couldn't with second. Have they taken Left and Right Hand limit? I tried but my answer again came as 1. Thanks in advance.
We have $\lim_{n \to \infty}x^n = 0$ if $|x|<1$.
$$\lim_{n \to \infty}\frac{x^{2n}-1}{x^{2n}+1}=\frac{0-1}{0+1}=-1$$
Note that $x$ here is fixed.
Hint:
What is \begin{equation*} \lim _{n\rightarrow \infty } a^{n} \end{equation*} when $-1<a<1$? Using this result, what is \begin{equation*} \lim _{n\rightarrow \infty }\frac{a^{n} -1}{a^{n} +1} \end{equation*} when $-1<a<1$? Isn't it \begin{equation*} \frac{\lim _{n\rightarrow \infty } a^{n} -1}{\lim _{n\rightarrow \infty } a^{n} +1} \end{equation*} Can you take it from here and complete the answer?
