Limits at Infinity, Part 3

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