# One-Sided Limits

How to find one-sided limits (left-hand limit and right-hand limit) and the limit of a function: examples and their solutions.

## Example 1

*x* → 1^{-} means*x* goes to 1

from the left side.

This is why when (-) is added,

the limit is called the 'left-hand limit'.

As *x* goes to 1^{-},

(= *x* goes to 1 from the left side.)*f*(*x*) goes to 2^{-}.

(= *f*(*x*) goes to 2 from downward.)

You don't have to write the (-)

when writing the answer.

So 2 is the answer.

*x* → 1^{+} means*x* goes to 1

from the right side.

This is why when (+) is added,

the limit is called the 'right-hand limit'.

As *x* goes to 1^{+},

(= *x* goes to 1 from the right side)*f*(*x*) goes to 2.

(No (+) or (-).)

The left-hand limit of *f*(*x*), 2,

and the right-hand limit *f*(*x*), 2,

are equal.

Then the limit of *f*(*x*) is that equal value: 2.

The limit of *f*(*x*) as *x* → *a*

has nothing to do with *f*(*a*).

See the given graph.

You can see that

the limit of *f*(*x*) as *x* → 1 is 2,

which has nothing to do with *f*(1): 4.

## Example 2

As *x* goes to 3^{-}

(= goes to 3 from the left side),*f*(*x*) goes to 2.

As *x* goes to 3^{+}

(= goes to 3 from the right side),*f*(*x*) goes to 3^{+}.

(= goes to 3 from upward.)

So 3 is the answer.

The left-hand limit of *f*(*x*), 2,

and the right-hand limit *f*(*x*), 3,

are not equal.

Then the limit of *f*(*x*) doesn't exist.

The limit of a function exist

if and only if

the left-hand limit and the right-hand limit are equal.