
Limits and Continuity 3-11
⇒ <
−
<
⇒
−
−
< − <
1
4
1 1
2
3 3
4 9
3
2
3
3
2
4 9x
x
x
x d.
So we set d = min {1/4, 2e/3} to get the required result
x
x4 9
1
−
− < e.
This completes the proof.
Example 3.9 Prove that
lim .
x
x
→
+
=
0
0
Solution: Let e > 0 be any real number.
Now, we need to find a d such that x − <0 e whenever 0 < ½x½ < 0 + d
i.e., whenever
i.e., whenever
i.e., wh
x x
x x
x
< < <
< < <
<
ε δ
ε δ
ε
0
0
2
eenever 0 < <x δ.
Therefore, we set d = e
2
to get the required result.
This completes the proof.
3.1.2 Properties of Limits
Now that we have defined formally what a limit is and examined several examples. We state and prove
some properties that are useful for working with ...