Question

Use the relation to determine the limit refer to picture please.

Answer 1

We solve as follows:

Since we have:

`\lim _(\theta\rightarrow0)\frac{2\sin (\sqrt[]{2}\theta)}{\sqrt[]{2}\theta}`

We can proceed as follows:

We assign a new denomination:

`\alpha=\sqrt[]{2}\theta`

And we solve for theta = 0:

`\alpha=\sqrt[]{2}(0)\Rightarrow\alpha=0`

Now, we re-write the original expression:

`\lim _(\alpha\rightarrow0)(2\sin(\alpha))/(\alpha)\Rightarrow2\lim _(\alpha\rightarrow0)(\sin(\alpha))/(\alpha)=2`

And, since we already know the following:

`\lim _(\theta\rightarrow0)(\sin (\theta))/(\theta)=1`

Then, we will have that the solution will be:

`\lim _(\theta\rightarrow0)\frac{2\sin (\sqrt[]{2}\theta)}{\sqrt[]{2}\theta}=2`