Can Continuity Ensure Infinite Limits in Nested Functions?

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In summary, the limit of a function f(x) as x approaches infinity is the value that f(x) gets closer and closer to as x gets larger and larger. To prove this, it must be shown that for any arbitrarily large number M, there exists a corresponding value of x (x > a) where f(x) is larger than M. Proving a limit as x approaches infinity is significant because it helps understand the behavior of a function as its input values get larger, and can identify asymptotes and overall shape and behavior. The limit of a function can approach infinity from both positive and negative directions, depending on the behavior of the function. There are other methods to prove this, such as the squeeze theorem and L'Hopital
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Euge
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Here is this week's POTW:

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Prove that if $f:\mathbf{R} \to \mathbf{R}$ is continuous and $\lim\limits_{x\to \infty} f(f(x)) = \infty$, then $\lim\limits_{x\to \infty} |f(x)| = \infty$.
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No one solved this problem. You can read my solution below.

If the conclusion is false, there is an $M > 0$ and a sequence $x_n \to \infty$ such that $|f(x_n)| \le M$ for all $n\in \mathbb{N}$. By the Bolzano - Weierstrass theorem, $\{f(x_n)\}$ has a convergent subsequence $\{f(x_{n_k})\}$. Suppose $f(x_{n_k}) \to L$. Continuity of $f$ implies $f(f(x_{n_k})) \to f(L)$. This contradicts the assumption $\lim\limits_{x\to \infty} f(f(x)) = \infty$.

Edit: Sorry, I've overlooked Opalg's solution, which you can read below!

Proof by contradiction (it involves knowing how to negate existential statements!).

Suppose that the result is false. Then there is a sequence $(x_n)$ such that $\lim\limits_{n\to\infty}x_n = \infty$ but $|f(x_n)|$ does not tend to infinity as $n\to\infty$. This means that there exists $R\in\mathbf{R}$ such that $|f(x_n)|\leqslant R$ infinitely often. So there is a subsequence $x_{n_k}$ (which satisfies $\lim\limits_{k\to\infty}x_{n_k} = \infty$) such that $|f(x_{n_k})| \leqslant R$ for all $k$.

The function $f$ is continuous, so it is bounded on closed bounded subsets of $\mathbf{R}$. Since the interval $[-R,R]$ is closed and bounded, it follows that there exists $M\in\mathbf{R}$ such that $f(x) \leqslant M$ whenever $|x|\leqslant R$.

Putting all that together, it follows that $f(f(x_{n_k})) \leqslant M$ for all $k$. But that contradicts the fact that $\lim\limits_{x\to\infty}f(f(x)) = \infty$.
 
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FAQ: Can Continuity Ensure Infinite Limits in Nested Functions?

What does it mean for a limit to approach infinity?

When a limit approaches infinity, it means that the function's output is growing without bound as the input approaches a certain value. In other words, the function's values are becoming larger and larger without a limit.

How do you prove that a limit approaches infinity?

In order to prove that a limit approaches infinity, you need to show that for any arbitrarily large number M, there exists a corresponding input value x such that the output of the function is greater than M. This can be done using the formal definition of a limit or through various limit comparison tests.

Can a limit approach infinity from both positive and negative directions?

Yes, a limit can approach infinity from both positive and negative directions. This means that as the input values get closer and closer to the limit, the function's output values will either get larger and larger (approaching positive infinity) or smaller and smaller (approaching negative infinity).

What types of functions typically have limits that approach infinity?

Functions that have limits that approach infinity are typically ones that have a horizontal asymptote at infinity, meaning that the function's output values approach infinity as the input values get larger and larger. Examples of these types of functions include exponential, logarithmic, and rational functions.

How does proving a limit approaches infinity relate to the behavior of a function?

Proving that a limit approaches infinity is important in understanding the behavior of a function at a certain point. It can help determine if a function has a vertical asymptote at that point, or if the function is increasing or decreasing without bound. This information is useful in graphing and analyzing the behavior of a function.

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