How do I find two-sided limits?

May 1, 2016

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}