For the limit, either evaluate immediately, or if it is indeterminate, find the form it represents, and then evaluate using the methods of this section.

lim x → 0 (1 + tan(2x))^(3/x)

Question

For the limit, either evaluate immediately, or if it is indeterminate, find the form it represents, and then evaluate using the methods of this section.
 
lim x → 0 (1 + tan(2x))^(3/x)

zugzwang · Answer

Well, it's indeterminate all right, just find which form?

anonymous · Answer

and then evaluate it
that's the hard part

anonymous · Answer

I know it's 0^0

zugzwang · Answer

$$\lim_{x \rightarrow 0}\left( 1 + \tan2x \right)^{\frac{3}{x}}$$
This is actually of the form
$$1^{\infty}$$

zugzwang · Answer

1 + tan 2x evaluated at x = 0 is just 1
3/x goes to infinity as x goes to 0.

zugzwang · Answer

Can you do it from here?

anonymous · Answer

and when I get 1/0 I can't use LR so what do I do?

zugzwang · Answer

Suppose
$$L = \lim_{x \rightarrow 0}\left( 1 + \tan2x \right)^{\frac{3}{x}}$$

zugzwang · Answer

Something doesn't want me to do this...

anonymous · Answer

What?

zugzwang · Answer

Hang on, I'll fix something up...

zugzwang · Answer

Ok, have a look at this