Worked examples
The question
Question: find the limit as x → 2 of (x² − 4)/(x − 2), using a table.
lim x → 2 x2 − 4x − 2 =
1 Try x = 2
First, try x = 2 itself.
The top is 0, and the bottom is 0.
22 − 42 − 2 = 00
We cannot divide by 0.
So f(x) has no value at 2.
00: no value
But a limit does not need the value at 2.
It asks: as x gets close to 2, what number does f(x) get close to?
limit: the number f(x) gets close to
Careful!
0/0 does not mean there is no limit.
It only means we must look closer.
Careful: not 00 → no limit ✗
2 From the left
Take numbers just below 2: 1.9, 1.99, 1.999.
They come closer and closer to 2.
Put 1.9 in: we get 3.9.
f(1.9) = (1.9)2 − 4(1.9) − 2 = 3.61 − 41.9 − 2 = −0.39−0.1 = 3.9
3.9
The others the same way, with a calculator.
3.99
3.999
From the left, f(x) gets close to 4.
from the left: f(x) → 4
3 From the right
Now numbers just above 2: 2.1, 2.01, 2.001.
Work out f(x) for each one.
4.1
4.01
4.001
From the right, f(x) gets close to 4 too.
from the right: f(x) → 4
4 Read the limit
Both sides get close to the same number.
So the limit is 4.
lim x → 2 x2 − 4x − 2 = 4
5 Check
Let's check with numbers even closer.
x = 1.9999 gives 3.9999.
x = 1.9999: f(x) = 3.9999
x = 2.0001 gives 4.0001.
Still close to 4.
Right!
x = 2.0001: f(x) = 4.0001
Let's look back.
First: try x = 2.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.
Answer lim x → 2 x2 − 4x − 2 = 4
▶ Watch it solved
The question
Question: find the limit as x → −1 of (x² − 1)/(x + 1), using a table.
lim x → −1 x2 − 1x + 1 =
1 Try x = −1
First, try x = −1 itself.
The top is 0, and the bottom is 0.
(−1)2 − 1(−1) + 1 = 00
We cannot divide by 0.
So f(x) has no value at −1.
00: no value
But a limit does not need the value at −1.
It asks: as x gets close to −1, what number does f(x) get close to?
limit: the number f(x) gets close to
Careful!
0/0 does not mean there is no limit.
It only means we must look closer.
Careful: not 00 → no limit ✗
2 From the left
Take numbers just below −1: −1.1, −1.01, −1.001.
They come closer and closer to −1.
Put −1.1 in: we get −2.1.
f(−1.1) = (−1.1)2 − 1(−1.1) + 1 = 1.21 − 1−1.1 + 1 = 0.21−0.1 = −2.1
−2.1
The others the same way, with a calculator.
−2.01
−2.001
From the left, f(x) gets close to −2.
from the left: f(x) → −2
3 From the right
Now numbers just above −1: −0.9, −0.99, −0.999.
Work out f(x) for each one.
−1.9
−1.99
−1.999
From the right, f(x) gets close to −2 too.
from the right: f(x) → −2
4 Read the limit
Both sides get close to the same number.
So the limit is −2.
lim x → −1 x2 − 1x + 1 = −2
5 Check
Let's check with numbers even closer.
x = −1.0001 gives −2.0001.
x = −1.0001: f(x) = −2.0001
x = −0.9999 gives −1.9999.
Still close to −2.
Right!
x = −0.9999: f(x) = −1.9999
Let's look back.
First: try x = −1.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.
Answer lim x → −1 x2 − 1x + 1 = −2
▶ Watch it solved
The question
Question: find the limit as x → −1 of (x² + 5x + 4)/(x + 1), using a table.
lim x → −1 x2 + 5x + 4x + 1 =
1 Try x = −1
First, try x = −1 itself.
The top is 0, and the bottom is 0.
(−1)2 + 5(−1) + 4(−1) + 1 = 00
We cannot divide by 0.
So f(x) has no value at −1.
00: no value
But a limit does not need the value at −1.
It asks: as x gets close to −1, what number does f(x) get close to?
limit: the number f(x) gets close to
Careful!
0/0 does not mean there is no limit.
It only means we must look closer.
Careful: not 00 → no limit ✗
2 From the left
Take numbers just below −1: −1.1, −1.01, −1.001.
They come closer and closer to −1.
Put −1.1 in: we get 2.9.
f(−1.1) = (−1.1)2 + 5(−1.1) + 4(−1.1) + 1 = 1.21 − 5.5 + 4−1.1 + 1 = −0.29−0.1 = 2.9
2.9
The others the same way, with a calculator.
2.99
2.999
From the left, f(x) gets close to 3.
from the left: f(x) → 3
3 From the right
Now numbers just above −1: −0.9, −0.99, −0.999.
Work out f(x) for each one.
3.1
3.01
3.001
From the right, f(x) gets close to 3 too.
from the right: f(x) → 3
4 Read the limit
Both sides get close to the same number.
So the limit is 3.
lim x → −1 x2 + 5x + 4x + 1 = 3
5 Check
Let's check with numbers even closer.
x = −1.0001 gives 2.9999.
x = −1.0001: f(x) = 2.9999
x = −0.9999 gives 3.0001.
Still close to 3.
Right!
x = −0.9999: f(x) = 3.0001
Let's look back.
First: try x = −1.
Then: from the left.
Then: from the right.
Then: read the limit.
Last: check.
Answer lim x → −1 x2 + 5x + 4x + 1 = 3
▶ Watch it solved
