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Suppose $f(x)$ is a function satisfying the inequalities:

$$2x-1\leq f(x)\leq x^4-x+3.$$

Based on this information, which of the following statements might be true about $f(x)$?

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

$\lim_{x\to 1}f(x)=2$

$\lim_{x\to 1}f(x)=3$

$\lim_{x\to 1}f(x)=4$

$\lim_{x\to 1}f(x)\; \text{does not exist}$