One-Sided Limits

First find the left-hand limit.

This is when x → 2^-:
x < 2.

So the left-hand limit of f(x) is
the limit of (-x + 6) as x → 2^-.

Put 2^- into (-x + 6).

Limit of a Function

2^- means
the number a little bit smaller than 2.
(almost the same)

So -(2^-) = -2.

-2 + 6 = 4

So the left-hand limit is 4.

Next, find the right-hand limit.

This is when x → 2⁺:
x > 2.

So the right-hand limit of f(x) is
the limit of x² as x → 2⁺.

Put 2⁺ into x².

2⁺ means
the number a little bit bigger than 2.
(almost the same)

So (2⁺)² = 2².

2² = 4

So the right-hand limit is 4.

The left-hand limit is 4.
The right-hand limit is 4.

So the left-hand limit and the right-hand limit
are equal: 4.

So the limit of f(x) exist: 4.

So 4 is the answer.

First find the left-hand limit.

This is when x → 0^-.

So the left-hand limit of f(x) is
the limit of x(x + 1)/|x| as x → 0^-.

This is when x → 0^-:
x < 0.

So change |x| to -x.

Absolute Value

Cancel the x factors.

Then x(x + 1)/(-x) = -(x + 1).

Put 0^- into -(x + 1).

0^- means
the number a little bit smaller than 0.
(almost the same)

So 0^- = 0.

-(0 + 1) = -1

So the left-hand limit is -1.

Next, find the right-hand limit.

This is when x → 0⁺.

So the right-hand limit of f(x) is
the limit of x(x + 1)/|x| as x → 0⁺.

This is when x → 0⁺:
x > 0.

So change |x| to x.

Absolute Value

Cancel the x factors.

Then x(x + 1)/x = x + 1.

Put 0⁺ into x + 1.

0⁺ means
the number a little bit bigger than 0.
(almost the same)

So 0⁺ = 0.

0 + 1 = 1

So the right-hand limit is 1.

The left-hand limit is -1.
The right-hand limit is 1.

So the left-hand limit and the right-hand limit
are not equal.

Then the limit of f(x) does not exist.

So the limit of f(x) does not exist.