calculus - How can I solve these trigonometric limits without using L'Hopital's Rule?

I don't understand why #1 is "does not exist" as opposed to 0. I also don't know how to begin solving #2 and #3. When I look it up, the only solutions involve L'Hopital's Rule but my teacher hasn't taught us it yet so I can't use it. Any help is appreciated!

Answer

For the first one the expression under the radical sign is negative for $\theta \ne \pi/2 $ so it is not real.

For the second one multiply top and bottom by $1-\cos \pi x$ and turn the top into $\sin^2 \pi x$

Then write the $\tan ^2 \pi x$of the bottom in terms of $ \sin \pi x$ and $\cos \pi x$ and cancel the $\sin ^2 \pi x$ from top and bottom.

The rest is easy.

The third one is similar to the second one.