Find the limit of composition 2

Calculus Level 3

Consider the function $f(x) = x^3$ . Let $f^{(n) } (x) = f\left( f^{(n-1)} (x) \right)$ be the function composed $n$ times.

Define the function $F: \mathbb{R} \rightarrow$ extended reals (including infinity) by

$F(x) = \lim_{n\rightarrow \infty} f^{(n)} (x).$

How many points of discontinuity does $F(x)$ have?

The answer is 2.

1 solution

Chew-Seong Cheong
May 25, 2016

We note that $f_n(x) = x^{3n}$ , therefore, we have:

$\displaystyle \lim_{n \to \infty} f_n (x) = \lim_{n \to \infty} x^{3n} = \begin{cases} - \infty & \text{if }x < -1 \\ -1 & \text{if }x = -1 \\ 0 & \text{if } - 1< x < 1 \\ 1 & \text{if }x = 1 \\ \infty & \text{if }x > 1 \end{cases}$

Yes thats the answer. But I solved it graphically.

Prince Loomba - 5 years ago

Prince, I edited the problem for you. Please learn up how to use LaTex. It is easy. You place your mouse cursors over the formulas to see the codes.

Chew-Seong Cheong - 5 years ago

Thanks for editting. Actually I have very less time to devote to brilliant. Since I am going to coaching as well as school. So I cant think of learning it this year. Maybe from next year, I will.

Prince Loomba - 5 years ago

@Prince Loomba – OK, but it won't take long to learn.

Chew-Seong Cheong - 5 years ago

@Chew-Seong Cheong – I have learnt some while posting problems. When I use laptop, I find math editor, so I post the problems in latex. But mostly I use my tablet

Prince Loomba - 5 years ago

@Prince Loomba – You can just key in the codes without the editor in Brilliant website. That is what I do. Sometimes with my phone too. I only refer to the editor only I need to key in new symbols or characters. Anyway, it is up to you.

Chew-Seong Cheong - 5 years ago

Find the limit of composition 2

The answer is 2.

1 solution

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