- The uniqueness theorem of the limit states that if $f(x)$ has a finite limit as $x$ approaches $x_0$, then that limit is unique
- The theorem is proven by assuming the limit is not unique, using contradiction, and applying the principles of sign preservation and comparison
- The sign preservation theorem explains that if the limit of $f(x)$ is $l$ (where $l$ is not equal to 0) and $f(x)$ and $l$ are both $\pm \infty$, then the sign of $f(x)$ is preserved
- The inverse theorem states that if $-f(x) > 0$, then $f(x) < 0$, and if $f(x) < 0$, then $-f(x) > 0$
- The two policemen theorem, also known as the comparison theorem, explains that if three functions $h(x)$, $f(x)$, and $g(x)$ are defined in the same neighborhood $H$ of $X$ and $h(x) ≤ f(x) ≤ g(x)$ in every point of $H$, and $lim h(x) = lim g(x) = l$, then $lim f(x) = l*.
- The uniqueness of the limit theorem is proven by contradiction and using the principles of sign preservation and comparison.
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