AP CALCULUS 1004 Limits pt.3 Limits at Infinity and End Behavior.
Limits at Infinity, Part 3
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Transcript of Limits at Infinity, Part 3
![Page 1: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/1.jpg)
![Page 2: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/2.jpg)
![Page 3: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/3.jpg)
Limits with RadicalsExample 1
![Page 4: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/4.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
![Page 5: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/5.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√x2 + 1
x + 1
![Page 6: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/6.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√x2 + 1
x + 1
Here we have a radical.
![Page 7: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/7.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√x2 + 1
x + 1
Here we have a radical.We’re going to do something similar to our basic technique.
![Page 8: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/8.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√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:
![Page 9: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/9.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√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:
limx→∞
√x2+1x2
.x2
x + 1
![Page 10: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/10.jpg)
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√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:
limx→∞
√x2+1x2
.x2
x + 1
limx→∞
√(1 + 1
x2
)x2
x + 1
![Page 11: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/11.jpg)
Limits with RadicalsExample 1
1. x < 0.
![Page 12: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/12.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
1. x < 0.
![Page 13: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/13.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.
![Page 14: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/14.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.
![Page 15: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/15.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.As we are taking a positive square root:
![Page 16: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/16.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.As we are taking a positive square root:
limx→∞
(−x)
√1 + 1
x2
x + 1
![Page 17: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/17.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.As we are taking a positive square root:
limx→∞
(−x)
√1 + 1
x2
x + 1
We can also write this as:
![Page 18: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/18.jpg)
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.As we are taking a positive square root:
limx→∞
(−x)
√1 + 1
x2
x + 1
We can also write this as:
− limx→∞
√1 + 1
x2
x+1x
![Page 19: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/19.jpg)
Limits with RadicalsExample 1
![Page 20: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/20.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
![Page 21: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/21.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
![Page 22: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/22.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
![Page 23: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/23.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
= − limx→∞
√1 + 1
x2
1 + 1x
![Page 24: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/24.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
= − limx→∞
√1 +��70
1x2
1 + 1x
![Page 25: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/25.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
= − limx→∞
√1 +��70
1x2
1 +���0
1x
![Page 26: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/26.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
= − limx→∞
√1 +��70
1x2
1 +���0
1x
= −√
1
1= −1
![Page 27: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/27.jpg)
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
= − limx→∞
√1 +��70
1x2
1 +���0
1x
= −√
1
1= −1
![Page 28: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/28.jpg)
Limits with RadicalsExample 1
2. x > 0.
![Page 29: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/29.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
![Page 30: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/30.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
![Page 31: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/31.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
![Page 32: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/32.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
![Page 33: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/33.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
![Page 34: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/34.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
= limx→∞
√1 + 1
x2
x+1x
![Page 35: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/35.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
= limx→∞
√1 + 1
x2
x+1x
= limx→∞
√1 + 1
x2
1 + 1x
![Page 36: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/36.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
= limx→∞
√1 + 1
x2
x+1x
= limx→∞
√1 +��70
1x2
1 + 1x
![Page 37: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/37.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
= limx→∞
√1 + 1
x2
x+1x
= limx→∞
√1 +��70
1x2
1 +���0
1x
![Page 38: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/38.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
= limx→∞
√1 + 1
x2
x+1x
= limx→∞
√1 +��70
1x2
1 +���0
1x
= 1
![Page 39: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/39.jpg)
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
limx→∞
(x)
√1 + 1
x2
x + 1
= limx→∞
√1 + 1
x2
x+1x
= limx→∞
√1 +��70
1x2
1 +���0
1x
= 1
![Page 40: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/40.jpg)
Limits with RadicalsExample 1
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
![Page 41: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/41.jpg)
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
![Page 42: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/42.jpg)
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
![Page 43: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/43.jpg)
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
![Page 44: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/44.jpg)
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
![Page 45: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/45.jpg)
Limits with RadicalsExample 2
![Page 46: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/46.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
![Page 47: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/47.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)
![Page 48: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/48.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)This limit is different, but easily solved.
![Page 49: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/49.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)This limit is different, but easily solved.We just need to divide and multiply by the conjugate:
![Page 50: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/50.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)This limit is different, but easily solved.We just need to divide and multiply by the conjugate:
limx→∞
(√x2 + 1−
√x2 − 1
).
√x2 + 1 +
√x2 − 1√
x2 + 1 +√x2 − 1
![Page 51: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/51.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)This limit is different, but easily solved.We just need to divide and multiply by the conjugate:
limx→∞
(√x2 + 1−
√x2 − 1
).
√x2 + 1 +
√x2 − 1√
x2 + 1 +√x2 − 1
Doing the algebra in the numerator:
![Page 52: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/52.jpg)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)This limit is different, but easily solved.We just need to divide and multiply by the conjugate:
limx→∞
(√x2 + 1−
√x2 − 1
).
√x2 + 1 +
√x2 − 1√
x2 + 1 +√x2 − 1
Doing the algebra in the numerator:
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
![Page 53: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/53.jpg)
Limits with RadicalsExample 2
![Page 54: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/54.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
![Page 55: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/55.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
x2 + 1− x2 + 1√x2 + 1 +
√x2 − 1
![Page 56: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/56.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
![Page 57: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/57.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞
2√x2 + 1 +
√x2 − 1
![Page 58: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/58.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞
2
�����:
∞√x2 + 1 +
√x2 − 1
![Page 59: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/59.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞
2
�����:
∞√x2 + 1 +���
��:∞√x2 − 1
![Page 60: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/60.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞
2
������
����:∞√
x2 + 1 +√x2 − 1
![Page 61: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/61.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞���
������
�:02√
x2 + 1 +√x2 − 1
![Page 62: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/62.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞���
������
�:02√
x2 + 1 +√x2 − 1
= 0
![Page 63: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/63.jpg)
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
= limx→∞���
������
�:02√
x2 + 1 +√x2 − 1
= 0
![Page 64: Limits at Infinity, Part 3](https://reader033.fdocuments.us/reader033/viewer/2022052218/547d1d31b37959932b8b529b/html5/thumbnails/64.jpg)