I need to determine if the following statements are true or false:
- If $f$ is a continuous function on the interval $[5,10)$ and $(x_n)$ is a convergent sequence in $[5,10)$ such that $\lim f(x_n) = 10^6$ then $\lim x_n$ can be determined.
- If $f$ is a continuous function on the interval $[5,10)$ and $(x_n)$ is a convergent sequence in $[5,10)$ such that $\lim f(x_n) = \infty$ then $\lim x_n$ can be determined.
Here are my arguments and questions:
If $x_n$ is convergent, then $\lim x_n \in [5,10]$ (the problem here is continuity at $x=10$). Is the author asking of the existence of limit of $(x_n)$ or if it can be uniquely determined? Because if $(x_n)$ is the sequence given by $x_n = 6-\frac{1}{n}$ and $f(x)= \frac{10^6}{6} x$ then $\lim f(x_n)= 10^6$. I could do the same for $x=7$ as well.
For the second question if $(x_n)$ converges to some point $c\in [5,10)$ then by the continuity of $f$, we must have $\lim f(x_n) = f(c) = \infty$. Thus, f is not continuous on $[5,10)$ and thus $\lim x_n$ cannot be determined. I'm not sure what to conclude when $c=10$.
I need some hints on this.
For the first part, you have exhibited two possible continuous functions of the form of
$$f_t(x) = \frac{10^6}{t}\cdot x$$ where $t \in \{ 6,7\}$ and a convergence sequence $x_{n,t}=t-\frac1n$ that shows that we can't determine the limit uniquely.
For the second part, note that for any $t \in [5, 10)$, $f(t) $ is finite. If $x_n$ converges to $t$, that is if $\lim_n x_n = t$, then we have $$\lim_{n \to \infty}f(x_n)= f(\lim_{n \to \infty} x_n ) = f(t)< \infty.$$
Hence, $\lim_{n \to \infty}x_n$ cannot be in $[5,10)$. It must converges to a limit point, the only limit point that is not in $[5,10)$ is $10$. Hence $\lim_{n \to \infty}x_n = 10$.