One-Sided Limits

Let $f : (a, b) \to R$ .

Let $a < c \leq b$ . We claim that we can define a left-handed limit (a limit from the left) as:

lim_{x \to c^{-}} f (x) = L

iff:

\forall (x_{n}) \subseteq (a, c) where x_{n} \to c, f (x_{n}) \to L

We may write

f (c^{-})

to denote this limit rather than

L

Similarly if $a \leq c < b$ , we write the right-handed limit (a limit from the right) as:

lim_{x \to c^{+}} f (x) = L

iff:

\forall (x_{n}) \subseteq (c, b) where x_{n} \to c, f (x_{n}) \to L

We may write

f (c^{+})

to denote this limit rather than

L

c \in (a, b)

here, then

lim_{x \to c} f (x)

exists iff both one-sided limits

lim_{x \to c^{-}} f (x), lim_{x \to c^{+}} f (x)

exist and are equal.

Proof

( $\Rightarrow$ ): If $lim_{x \to c} f (x) = L$ exists then by the Characterization of Continuity then both the sequences we can construct for $(x_{n}) \subseteq (a, c)$ and $(y_{n}) \subseteq (c, b)$ where $x_{n}, y_{n} \to L$ then $f (x_{n}), f (y_{n}) \to f (c)$ , thus showing both one-sided limits exist.

( $\Leftarrow$ ): Suppose both sided limits equal and exist. Let $(x_{n}) \subseteq (a, c)$ where $x_{n} \to c$ then by the left, sided limit existing, then $f (x_{n}) \to f (c)$ . Similarly if $(y_{n}) \subseteq (c, b)$ where $y_{n} \to c$ then $f (y_{n}) \to f (c)$ . This implies $lim_{x \to c^{-}} f (x) = lim_{x \to c^{+}} f (x) = f (c)$ so then the limit must exist by the Existence of Functional Limits.

#todo/school : Show the reverse implication 📅 2024-11-12 ✅ 2024-11-14

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