Question

How do I find two-sided limits?

Answer 1

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

Explanation:

For example,

as `x rarr0` through positive values `csc x to +oo`.

It `to-oo`, for approach through negative values.

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

1st and 3rd quadrants.

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

Definitions:

Limit through lower values is

lim h `rarr` 0 of f(a-h).

Limit through higher values is

lim h `rarr` 0 of f(a+h).

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

a = 0 and h = x.

Now, consider lim `x rarr` 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 `rarr 0` through negative

values. And so, the limit `-oo`.

Likewise, the right limit is `+oo`.

Here, the side limits `+- oo` and f(0) do not exist.

Another vivid example is `lim rarr 0` of `( abs x)/x`.

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

Here, the side limits are obviously `+-`1 and and f(0) does not

exist.

graph{(abs x)/x}