Stolz-Césaro Theorem: let $\{x_n\}, \{y_n\}$ be real sequences.
$\ \ \ \ $ $\infty/\infty$ case: if $y_n$ is monotone and unbounded, then (as shown here)
$$\liminf_{n\to\infty} \frac{\Delta x_n}{\Delta y_n} \le \liminf_{n\to\infty} \frac{x_n}{y_n} \le \limsup_{n\to\infty} \frac{x_n}{y_n} \le \limsup_{n\to\infty} \frac{\Delta x_n}{\Delta y_n}.$$
$\ \ \ \ $ $0/0$ case: if $x_n, y_n\to 0$, then we have (as shown here) that
$$\frac{\Delta x_n}{\Delta y_n} \to L \ \ \Rightarrow \ \ \frac{x_n}{y_n}\to L$$
(where the statement "$\Delta x_n/\Delta y_n \to L$" may be false if the limit doesn't exist.)
Observe how the statement for the $0/0$ case is weaker. It is natural to wonder if the following is true:
$\ \ \ \ $ Stronger $0/0$ case: given the real sequences $\{x_n\}, \{y_n\}$ so that $x_n,y_n\to 0$, we have that $$\liminf_{n\to\infty} \frac{\Delta x_n}{\Delta y_n} \le \liminf_{n\to\infty} \frac{x_n}{y_n} \le \limsup_{n\to\infty} \frac{x_n}{y_n} \le \limsup_{n\to\infty} \frac{\Delta x_n}{\Delta y_n}.$$
Is the statement above true? If so, how could one prove it? If not, what is a counter-example?
