Conceptual Doubt or Ambiguity in Limits

Question

I came across this question in my book. $$ Evaluate:~~~\lim:\lim_{n\to\infty}\frac{a^n+b^n}{a^n-b^n} $$ $$ where~~~~~a>b>0 $$ According to me this question should have 4 answers

Since for any $x$ $<$ $1$ we get $x^\infty$ $=$ $0$

And for any $x$ > $1$ we get $x^\infty$ $=$ $\infty$

Hence when $a>1$ and $b<1$ we get $\frac{\infty}{\infty}$ form

When $a<1$ and $b>1$ we get ($-$$\frac{\infty}{\infty}$) form

Similary when both $a$ and $b$ are greater than $1$ then we get a new form (I never encountered this form and I even don't know if it's indeterminate or not) $\frac{\infty}{\infty-\infty}$

When both $a$ and $b$ are $<$ $1$ we get $\frac{0}{0}$ form.

So 4 different forms hence we must get 4 different answers.

But my book says that the answer is $1$.

I am really confused and I even don't find anything wrong in my reasoning.

Could someone tell me what am I missing or what I did wrong.

Thankyou

Edit : After looking at the comments, I checked the question again and the condition given is, $a>b>0$ and not $a,b>0$ so second case is ruled out. But still 3 cases are there.

Edit 2 : This is a general case and if any number was assigned to $a$ or $b$ for example $a=3$ and $b=2$ then I would have found the limit easily but here it is not specified and only $a>b>0$ is given, so I get 3 different cases. Also they have not specified the difference like, $a>b>0$ then $a$ can be $>$ 1 but $b$ might be $<1$ and $a$ might be so large that $a\to\infty$ or the difference might be so small that $a≈b$, So I find some ambiguities here and I've also mentioned them in the cases, so someone please explain me what am I doing wrong.

Edit 3 : I saw the answer of Kavi Rama Murthy, $$ \lim:\lim_{n\to\infty}\frac {(\frac a b)^{n}+1}{(\frac ab )^{n}-1}=1 $$ So, what if $a≈b$ then the answer should be $\frac{2}{\to0}$ which is $\infty$ but if $a>>>>b$ then again indeterminate form $\frac{\infty}{\infty}$

Edit 4 : I think I got my answer, since $a>b>0$ Divide both sides by $a$ and now $\frac{a}{a}>\frac{b}{a}>\frac{0}{a}$ which gives $1>\frac{b}{a}>0$ and hence the answer is 1.

Answer 1

Divide numerator and denominator by $b^{n}$.

$\lim \frac {(\frac a b)^{n}+1}{(\frac ab )^{n}-1}=1$ if $a>b$, $-1$ if $ a<b$ and the expression is not defined when $a=b$.

Answer 2

The main point here is that the "forms" you are mentioning are indeterminate ones !

For instance : $\frac{n^2}{n} \rightarrow +\infty$, but $\frac{n}{n^2} \rightarrow 0$. And both are of the form "$\frac{\infty}{\infty}$".

When you are faced with these indeterminate forms, you may not compute the limit directly from these forms, as the example above shows. So, there is no contradiction.

In your example, the point is to understand which term is "dominant" between $a^n$ and $b^n$. In this case, since $a > b > 0$, the dominant term is $a^n$.

EDIT : Usually, when there are indeterminate forms, it means that you have to understand more precisely the asymptotic behavior of the terms involved.