# Two-Sided Limits

## Key Questions

For 2-sided limits about x = a, approach 'a' through higher and lower values, respectively.

#### Explanation:

For example,

as $x \rightarrow 0$ through positive values $\csc x \to + \infty$.

It $\to - \infty$, for approach through negative values.

See graph, close to y-axis, in both directions. respectively, in the

graph{y-1/sin x = 0[-10 10 -10 10] }

Definitions:

Limit through lower values is

lim h $\rightarrow$ 0 of f(a-h).

Limit through higher values is

lim h $\rightarrow$ 0 of f(a+h).

Here, f(x) = 1 / sin x and

a = 0 and h = x.

Now, consider lim $x \rightarrow$ 0_ of 1/sin x

For any x < 0 and close to 0, sin x is negative.

So, the limit is 1 / 0_, where 0_ means $\rightarrow 0$ through negative

values. And so, the limit $- \infty$.

Likewise, the right limit is $+ \infty$.

Here, the side limits $\pm \infty$ and f(0) do not exist.

Another vivid example is $\lim \rightarrow 0$ of $\frac{\left\mid x \right\mid}{x}$.

See the graph below. Observe that f(0) is indeterminate.

Here, the side limits are obviously $\pm$1 and and f(0) does not

exist.

graph{(abs x)/x}

IF L=lim_(x→A−)f(x) exists

AND R=lim_(x→A+)f(x) exists

AND $L = R$

THEN value $L = R$ is called a two-sided limit.

#### Explanation:

We are talking here about a limit of a function $f \left(x\right)$ as its argument $x$ approaches a concrete real number $A$ within its domain.
It's not a limit when an argument tends to infinity.

The argument $x$ can tend to a concrete real number $A$ in several ways:
(a) $x \to A$ while $x < A$, denoted sometimes as $x \to {A}^{-}$
(b) $x \to A$ while $x > A$, denoted sometimes as $x \to {A}^{+}$
(c) $x \to A$ without any additional conditions

All the above cases are different and conditional limits (a) and (b), when $x \to {A}^{-}$ and $x \to {A}^{+}$, might or might not exist independently from each other and, if both exist, might or might not be equal to each other.

Of course, if unconditional limit (c) of a function when $x \to A$ exists, the other two, the conditional ones, exist as well and are equal to the unconditional one.

The limit of $f \left(x\right)$ when $x \to {A}^{-}$ is a one-sided (left-sided) limit.
The limit of $f \left(x\right)$ when $x \to {A}^{+}$ is also one-sided (right-sided) limit.
If they both exist and equal, we can talk about two-sided limit

So, two-sided limit can be defined as follows:
IF
$L = {\lim}_{x \to {A}^{-}} f \left(x\right)$ exists AND
$R = {\lim}_{x \to {A}^{+}} f \left(x\right)$ exists AND
$L = R$
THEN value $L = R$ is called a two-sided limit.