1 votes 1 votes Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f^n(x)$ exists for every $x \in \mathbb{R}$, where $f^n(x) = f \circ f^{n-1}(x)$ for $n \geq 2$. Define $$S=\left\{\lim _{n \rightarrow \infty} f^n(x): x \in \mathbb{R}\right\} \text{ and } T=\left\{x \in \mathbb{R}:f(x)=x\right\}$$ Then which of the following is necessarily true? $S \subset T$ $T \subset S$ $S = T$ None of the above Calculus isi2019-mma engineering-mathematics calculus limits + – Sayan Bose asked May 7, 2019 edited Jul 17, 2021 by Arjun Sayan Bose 1.6k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
1 votes 1 votes Let $h(x) = \lim_{n \rightarrow \infty} f^n(x) = f(\lim_{n \rightarrow \infty}f^{n-1}(x))$ We can perform the last step as the function is continuous for every x. Continuing on, $h(x) = f(h(x))$ $ \forall x \in R$ because $n\rightarrow \infty\implies n-1 \rightarrow \infty$ It means $S = \{f(x) = x : \forall x \in Range(h(x))\}$ The function defined by two sets in same but $S$ is defined over some elements of real numbers whereas $T$ is defined for all real numbers. $\implies S \subset T$ $A$ is correct answer. pratekag answered May 7, 2019 pratekag comment Share Follow See all 4 Comments See all 4 4 Comments reply srestha commented May 7, 2019 reply Follow Share How do u know $T$ mapped on all real numbers. It also can be mapped on some real numbers 0 votes 0 votes pratekag commented May 7, 2019 reply Follow Share Given in question. f is defined on all real numbers because domain is R . and most importantly in the definition of T, $ x\in R$ which means no restriction. And this relation holds necessarily. But it could be the case that both S and T are equal if $Range(h(x)) = R$ , not otherwise. 0 votes 0 votes NastyBall commented Jul 9, 2021 reply Follow Share Your answer is Partially correct but wrong as per the official key. The correct answer is option $C$. 0 votes 0 votes pratekag commented Apr 30, 2022 reply Follow Share you are right. C is the correct answer. A function by definition is defined over all the elements of the domain. Otherwise it is not a function. 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes $ f^{n}(x)=f \circ f^{n-1}(x) $ $taking$ n = 2 , $ f^{2}(x)=f(f(x)) $ $ accordingly,f^{3}(x)=f(f(f(x))) $ putting the value of $f(x)=x$ S={f(x), where $x \in \mathbb{R}$} So T is a subset of S *Correct me if I am wrong Amartya answered Jun 11, 2020 edited Jun 11, 2020 by Amartya Amartya comment Share Follow See 1 comment See all 1 1 comment reply NastyBall commented Jul 9, 2021 reply Follow Share Your answer is definitely wrong. It should be Option $C$. 0 votes 0 votes Please log in or register to add a comment.