We solve as follows:
Since we have:
[tex]\lim _{\theta\rightarrow0}\frac{2\sin (\sqrt[]{2}\theta)}{\sqrt[]{2}\theta}[/tex]
We can proceed as follows:
We assign a new denomination:
[tex]\alpha=\sqrt[]{2}\theta[/tex]
And we solve for theta = 0:
[tex]\alpha=\sqrt[]{2}(0)\Rightarrow\alpha=0[/tex]
Now, we re-write the original expression:
[tex]\lim _{\alpha\rightarrow0}\frac{2\sin(\alpha)}{\alpha}\Rightarrow2\lim _{\alpha\rightarrow0}\frac{\sin(\alpha)}{\alpha}=2[/tex]
And, since we already know the following:
[tex]\lim _{\theta\rightarrow0}\frac{\sin (\theta)}{\theta}=1[/tex]
Then, we will have that the solution will be:
[tex]\lim _{\theta\rightarrow0}\frac{2\sin (\sqrt[]{2}\theta)}{\sqrt[]{2}\theta}=2[/tex]
