The function f is defined by f(x) = x^2 for x < 2 and f(x) = 3x

Step by step

The question

Question: find the limit as x → 2 of this f(x): x² if x < 2, and 3x − 2 if x ≥ 2.

f(x) = x² if x < 2

There are four choices.

First we solve it, the normal way.

Then we pick.

3x − 2 if x ≥ 2

1 From the left

From the left means x is a bit less than 2.

There, f(x) is x².

lim x → 2⁻ f(x) = lim x → 2⁻ x²

So we use the first piece.

x < 2: piece 1

A polynomial piece has no gaps, so we just put 2 in.

We get 4.

= 2² = 4

2 From the right

From the right means x is a bit more than 2.

There, f(x) is 3x − 2.

lim x → 2⁺ f(x) = lim x → 2⁺ (3x − 2)

So we use the second piece.

x > 2: piece 2

A polynomial piece has no gaps, so we just put 2 in.

We get 4.

= 3(2) − 2 = 6 − 2 = 4

3 Compare the sides

From the left we get 4.

From the right we get 4.

left: 4 right: 4

Both sides give the same number.

So the limit is 4.

lim x → 2 f(x) = 4

Careful!

One side is not enough.

A limit needs both sides to give the same number.

Careful: both sides must give the same number

4 Check

Let's check on the graph.

The first piece comes in from the left.

It reaches height 4.

The second piece starts at height 4.

They meet: no jump.

At x = 2 both pieces arrive at 4.

Right!

5 Pick the choice

Our answer is limit = 4.

That is choice B.

B) 4

6 Why not the others

A is 2.

Wrong: used x, not x².

A) 2 ✗ used x, not x^2

C is 8.

Wrong: added the two sides.

C) 8 ✗ added the two sides

D is 6.

Wrong: used 3x only and lost the − 2.

D) 6 ✗ used 3x only and lost the − 2

Let's look back.

First: from the left.

Then: from the right.

Then: compare the sides.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer