The function f is defined by f(x) = x^2 + 1 for x < 1 and f(x) = 3 -… Step by Step

Step by step

The question

Question: find the limit as x → 1 of this f(x): x² + 1 if x < 1, and 3 − x if x ≥ 1.

f(x) = x² + 1 if x < 1

There are four choices.

First we solve it, the normal way.

Then we pick.

3 − x if x ≥ 1

1 From the left

From the left means x is a bit less than 1.

There, f(x) is x² + 1.

lim x → 1⁻ f(x) = lim x → 1⁻ (x² + 1)

So we use the first piece.

x < 1: piece 1

A polynomial piece has no gaps, so we just put 1 in.

We get 2.

= 1² + 1 = 1 + 1 = 2

2 From the right

From the right means x is a bit more than 1.

There, f(x) is 3 − x.

lim x → 1⁺ f(x) = lim x → 1⁺ (3 − x)

So we use the second piece.

x > 1: piece 2

A polynomial piece has no gaps, so we just put 1 in.

We get 2.

= 3 − 1 = 2

3 Compare the sides

From the left we get 2.

From the right we get 2.

left: 2 right: 2

Both sides give the same number.

So the limit is 2.

lim x → 1 f(x) = 2

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 2.

The second piece starts at height 2.

They meet: no jump.

At x = 1 both pieces arrive at 2.

Right!

5 Pick the choice

Our answer is limit = 2.

That is choice D.

D) 2

6 Why not the others

A is 1.

Wrong: used x² only and lost the + 1.

A) 1 ✗ used x^2 only and lost the + 1

B is 4.

Wrong: added the two sides.

B) 4 ✗ added the two sides

C is 3.

Wrong: used 3 only and lost the − x.

C) 3 ✗ used 3 only and lost the − x

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