We solve limits by rationalizing. This is the second technique you may learn after limits by factoring. We solve two examples step by step. Watch video: http://www.youtube.com/watch?v=8CtpuojMJzA More videos and lessons: http://www.intuitive-calculus.com/solving-limits.html

3. Example 1

4. Example 1
Let’s try to find the limit:

5. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h

6. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.

7. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.

8. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
=

9. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0

10. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.

11. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a

12. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
¨
¨¨¨
¨¨¨B1√
a + h +
√
a
√
a + h +
√
a

13. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a

14. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a

15. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
a + h − a
h
√
a + h +
√
a

16. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a

17. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a
= lim
h→0
h
h
√
a + h +
√
a

18. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a
= lim
h→0
¡h
¡h
√
a + h +
√
a

19. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a
= lim
h→0
¡h
¡h
√
a + h +
√
a
= lim
h→0
1
√
a + h +
√
a

20. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a
= lim
h→0
¡h
¡h
√
a + h +
√
a
= lim
h→0
1
$$$$X
√
a√
a + h +
√
a

21. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a
= lim
h→0
¡h
¡h
√
a + h +
√
a
= lim
h→0
1
$$$$X
√
a√
a + h +
√
a
=
1
2
√
a

22. Example 1
Let’s try to find the limit:
lim
h→0
√
a + h −
√
a
h
We rationalize the expression.
In this case by multiplying and dividing by the conjugate.
lim
h→0
√
a + h −
√
a
h
= lim
h→0
√
a + h −
√
a
h
.
√
a + h +
√
a
√
a + h +
√
a
= lim
h→0
√
a + h
2
−
√
a
2
h
√
a + h +
√
a
= lim
h→0
¡a + h − ¡a
h
√
a + h +
√
a
= lim
h→0
¡h
¡h
√
a + h +
√
a
= lim
h→0
1
$$$$X
√
a√
a + h +
√
a
=
1
2
√
a

23. Example 2

24. Example 2
lim
x→0
√
1 + x − 1
x

25. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.

26. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
=

27. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.

28. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1

29. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
¨¨
¨¨¨
¨¨B1√
1 + x + 1
√
1 + x + 1

30. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1

31. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1

32. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
1 + x − 1
x
√
1 + x + 1

33. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1

34. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1
= lim
x→0
x
x
√
1 + x + 1

35. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1
= lim
x→0
&x
&x
√
1 + x + 1

36. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1
= lim
x→0
&x
&x
√
1 + x + 1
= lim
x→0
1
√
1 + x + 1

37. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1
= lim
x→0
&x
&x
√
1 + x + 1
= lim
x→0
1
$$$$$X
√
1√
1 + x + 1

38. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1
= lim
x→0
&x
&x
√
1 + x + 1
= lim
x→0
1
$$$$$X
√
1√
1 + x + 1
=
1
2

39. Example 2
lim
x→0
√
1 + x − 1
x
We rationalize using the same method.
lim
x→0
√
1 + x − 1
x
= lim
x→0
√
1 + x − 1
x
.
√
1 + x + 1
√
1 + x + 1
= lim
x→0
√
1 + x
2
− 12
x
√
1 + x + 1
= lim
x→0
¡1 + x − ¡1
x
√
1 + x + 1
= lim
x→0
&x
&x
√
1 + x + 1
= lim
x→0
1
$$$$$X
√
1√
1 + x + 1
=
1
2