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In this video we talk about limits at infinity with radicals. We solve two hairy problems.
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Limits with RadicalsExample 1
Limits with RadicalsExample 1
Let’s consider the limit:
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√x2 + 1
x + 1
Limits with RadicalsExample 1
Let’s consider the limit:
limx→∞
√x2 + 1
x + 1
Here we have a radical.
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.
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:
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
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
Limits with RadicalsExample 1
1. x < 0.
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
1. x < 0.
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.
Limits with RadicalsExample 1
limx→∞
√(1 + 1
x2
)x2
x + 1
Here we have two cases to consider:
1. x < 0.
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:
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
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:
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
Limits with RadicalsExample 1
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Limits with RadicalsExample 1
− limx→∞
√1 + 1
x2
x+1x
= − limx→∞
√1 + 1
x2
1 + 1x
Now, as we take the limit when x → +∞:
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
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
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
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
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
Limits with RadicalsExample 1
2. x > 0.
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
Limits with RadicalsExample 1
Now the second case:
2. x > 0.
limx→∞
√(1 + 1
x2
)x2
x + 1
We take the positive square root:
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
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
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
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
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
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
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
Limits with RadicalsExample 1
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
Limits with RadicalsExample 1
So, the answer is:
I The limit is 1 when x → +∞.
I The limit is −1 when x → −∞.
Limits with RadicalsExample 2
Limits with RadicalsExample 2
Let’s now consider the limit:
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)
Limits with RadicalsExample 2
Let’s now consider the limit:
limx→∞
(√x2 + 1−
√x2 − 1
)This limit is different, but easily solved.
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:
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
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:
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
Limits with RadicalsExample 2
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
x2 + 1− x2 + 1√x2 + 1 +
√x2 − 1
Limits with RadicalsExample 2
limx→∞
(√x2 + 1
)2−(√
x2 − 1)2
√x2 + 1 +
√x2 − 1
limx→∞
��x2 + 1−��x2 + 1√
x2 + 1 +√x2 − 1
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
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
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
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
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
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
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
