Continuous

Find the left-hand limit of f(x) at x = 1.

If x < 1,
f(x) = x² + 1.

Piecewise Function: Graph

So the left-hand limit is
the limit of (x² + 1) as x → 1^-.

Find the limit value.

Put 1^-
into (x² + 1).

Limit of a Function

(1^-)² = 1²

1 + 1 = 2

So the left-hand limit is 2.

Find the right-hand limit of f(x) at x = 1.

If x ≥ 1,
f(x) = -x + 3.

So the right-hand limit is
the limit of (-x + 3) as x → 1⁺.

Find the limit value.

Put 1⁺
into (-x + 3).

-(1⁺) = -1

-1 + 3 = 2

So the right-hand limit is 2.

Find the function value f(1).

If x ≥ 1,
f(x) = -x + 3.

So f(1) = -1 + 3 = 2.

The left-hand limit is 2.
The right-hand limit is 2.
And the function value f(1) is 2.

So the left-hand limit, the right-hand limit,
and the function value are all equal
at x = 1.

Then y = f(x) is continuous at x = 1.

So y = f(x) is continuous at x = 1.

Find the limit of f(x) at x = 3.

If x ≠ 3,
f(x) = (x + 1)(x - 3)/(x - 3).

So the limit value is
the limit of (x + 1)(x - 3)/(x - 3) as x → 3.

Cancel (x - 3) factors.

Find the limit value.

Put 3
into (x + 1).

Then 3 + 1 = 4.

So the limit value is 4.

Find the function value f(3).

If x = 3,
f(x) = 2.

So f(3) = 2.

The limit value is 4.
And the function value f(3) is 2.

So the limit value and the function value
are not equal
at x = 3.

Then y = f(x) is not continuous at x = 3.

So y = f(x) is not continuous at x = 3.