specifying an asymptotic function

Question

Please help me out;

I need to specify a function satisfying these conditions:

$$ f(0)=1 \;\;;\;\lim_{x \to \infty}f(x)=0$$

Hopefully does there exist a simple answer? Thanks a lot!

Answer 1

$$f(x) = \begin{cases}1 & \text{ if } x\leq 0 \\ 0&\text{ if }x>0\end{cases}$$

is one such function. Also, for any $\alpha > 0$, the function

$$\frac 1{(x + 1)^\alpha}$$

satisfies your conditions.

Furthermore, take any function $f(x)$ which is not identically equal to $0$ and is defined on $(a, \infty)$ for some $a\in\mathbb R$. Also, assume that $$\lim_{x\to\infty} f(x) = 0.$$Then there exists some $x_0$ such that $f(x_0)\neq 0$. In that case, the function

$$g(x) = \frac{f(x+x_0)}{f(x_0)}$$ Has the following properties:

$$\lim_{x\to\infty} g(x) = \lim_{x\to\infty} \frac{f(x+x_0)}{f(x_0)} = 0\\ g(0) = \frac{f(x_0)}{f(x_0)} = 1$$

Answer 2

There are many trillions of answers. The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 0$ except for $f(0) = 1$ will do.