A function with a finite limit at (both) 0 and infinity

Question

I ask myself if there exists a $f(x)$ function that limit goes to a finite number for both when x goes to 0 and to infinity. Is it possible in some way ?

Answer 1

How about the constant function $f(x)=1$? $$\lim_{x\to 0}f(x)=1\qquad \lim_{x\to\infty}f(x)=1$$

Answer 2

Not only do such functions exist, there are a whole lot of them!

• Constant functions, $f(x) = C$, satisfy your condition
• Continuous functions with limits at infinity satisfy your condition, i.e. $e^{-x^2}$, or $\frac{1}{1+x^2}$ or $\arctan(x)$.
• Let's take a $\delta > 0$ and $M>\delta>0$. Then, take any continuous funtion $f:[0, \delta)$ a function $g:(M, \infty)\to \mathbb R$ for which the limit $$\lim_{x\to\infty} g(x)$$ exists, and one arbitrary function $h:[\delta, M]$. Then the function $$F(x)\begin{cases} f(x) & x < \delta\\ h(x) & \delta\leq x\leq M\\ g(x) & M<x \end{cases}$$ has a limit as $x$ approaches $0$ and $\infty$.