The document provides two examples of limits involving radicals. In the first example, the limit is evaluated as x approaches positive infinity and negative infinity. The limit is 1 as x approaches positive infinity, and -1 as x approaches negative infinity. In the second example, the limit is simplified by dividing and multiplying by the conjugate, and the limit is evaluated to be 0 as x approaches infinity.

3. Limits with Radicals
Example 1

4. Limits with Radicals
Example 1
Let’s consider the limit:

5. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1

6. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.

7. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.

8. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
So, we divide and multiply by x2 inside the radical symbol:

9. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
So, we divide and multiply by x2 inside the radical symbol:
lim
x→∞
x2+1
x2 .x2
x + 1

10. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
So, we divide and multiply by x2 inside the radical symbol:
lim
x→∞
x2+1
x2 .x2
x + 1
lim
x→∞
1 + 1
x2 x2
x + 1

11. Limits with Radicals
Example 1

12. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1

13. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:

14. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.

15. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:

16. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
lim
x→∞
(−x)
1 + 1
x2
x + 1

17. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
lim
x→∞
(−x)
1 + 1
x2
x + 1
We can also write this as:

18. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
lim
x→∞
(−x)
1 + 1
x2
x + 1
We can also write this as:
− lim
x→∞
1 + 1
x2
x+1
x

19. Limits with Radicals
Example 1

20. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x

21. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x

22. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:

23. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 + 1
x2
1 + 1
x

24. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 + 1
x

25. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x

26. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= −
√
1
1
= −1

27. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= −
√
1
1
= −1

28. Limits with Radicals
Example 1

29. Limits with Radicals
Example 1
Now the second case:

30. Limits with Radicals
Example 1
Now the second case:
2. x 0.

31. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1

32. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:

33. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1

34. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x

35. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 + 1
x2
1 + 1
x

36. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 + 1
x

37. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x

38. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= 1

39. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= 1

40. Limits with Radicals
Example 1

41. Limits with Radicals
Example 1
So, the answer is:

42. Limits with Radicals
Example 1
So, the answer is:
The limit is 1 when x → +∞.

43. Limits with Radicals
Example 1
So, the answer is:
The limit is 1 when x → +∞.
The limit is −1 when x → −∞.

44. Limits with Radicals
Example 1
So, the answer is:
The limit is 1 when x → +∞.
The limit is −1 when x → −∞.

45. Limits with Radicals
Example 2

46. Limits with Radicals
Example 2
Let’s now consider the limit:

47. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1

48. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.

49. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:

50. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
lim
x→∞
x2 + 1 − x2 − 1 .
√
x2 + 1 +
√
x2 − 1
√
x2 + 1 +
√
x2 − 1

51. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
lim
x→∞
x2 + 1 − x2 − 1 .
√
x2 + 1 +
√
x2 − 1
√
x2 + 1 +
√
x2 − 1
Doing the algebra in the numerator:

52. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
lim
x→∞
x2 + 1 − x2 − 1 .
√
x2 + 1 +
√
x2 − 1
√
x2 + 1 +
√
x2 − 1
Doing the algebra in the numerator:
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1

53. Limits with Radicals
Example 2

54. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1

55. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1

56. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1

57. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞
2
√
x2 + 1 +
√
x2 − 1

58. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞
2
$$$$$X∞√
x2 + 1 +
√
x2 − 1

59. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞
2
$$$$$X∞√
x2 + 1 +$$$$$X∞√
x2 − 1

60. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞
2
$$$$$$$$$$X∞
√
x2 + 1 +
√
x2 − 1

61. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞$$$$$$$$$$X0
2
√
x2 + 1 +
√
x2 − 1

62. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞$$$$$$$$$$X0
2
√
x2 + 1 +
√
x2 − 1
= 0

63. Limits with Radicals
Example 2
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1
lim
x→∞
x2 + 1 − x2 + 1
√
x2 + 1 +
√
x2 − 1
= lim
x→∞$$$$$$$$$$X0
2
√
x2 + 1 +
√
x2 − 1
= 0