How to find the limit of this function?

[bleak heron]

Have I done it correctly?

[true hinge]

I assume you meant 1/(2x) in the power.
For x -> 0 we have tan(x) ~ x. So:
(1 + tan(√(3x))^2)^(1/(2x)) ~ (1 + √(3x)^2)^(1/(2x)) = (1 + 3x)^(1/(2x)), x -> 0
The rest is easy, just recall what the limit of (1 + x)^(1/x) for x -> 0 is.

[bleak heron]

Oh btw how did you assume that tanx ~ x?

[true hinge]

I didn't assume it, it's a known thing.
Since tan(x) = x + O(x^3) for x -> 0, that means tan(x) ~ x for x -> 0.

[bleak heron]

Ahh did it come from the known identity that lim x->0 tanx/x=1 btw?