# How do I find two-sided limits?

##### 1 Answer

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

#### Explanation:

For example,

as

It

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

Limit through higher values is

lim h

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

a = 0 and h = x.

Now, consider lim

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

So, the limit is 1 / 0_, where 0_ means

values. And so, the limit

Likewise, the right limit is

Here, the side limits

Another vivid example is

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

Here, the side limits are obviously

exist.

graph{(abs x)/x}