Find the limit of the sequence

find the limit of the sequence whose terms are given by:

an = [(1/(e^(4n) + n^2))]^(1/n)

We have:
lim (n-->infinity) a_n
= lim (n-->infinity) {1/[e^(4n) + n^2]}^(1/n)
= lim (n-->infinity) 1/[e^(4n) + n^2]/^(1/n)
= 1/{lim (n-->infinity) [e^(4n) + n^2]^(1/n)}.

Then, if we let:
L = lim (n-->infinity) [e^(4n) + n^2]^(1/n),

and take the natural log of both sides, we get:
ln(L) = lim (n-->infinity) ln{[e^(4n) + n^2]^(1/n)}
= lim (n-->infinity) {(1/n)ln[e^(4n) + n^2]}
= lim (n-->infinity) ln[e^(4n) + n^2]/n.

Then, by applying L'Hopital's Rule:
ln(L) = lim (n-->infinity) ln[e^(4n) + n^2]/n
= lim (n-->infinity) [4e^(4n) + 2n]/[e^(4n) + n^2]
= 4.

So, since ln(L) = 4:
L = lim (n-->infinity) [e^(4n) + n^2]^(1/n) = e^4.

Finally, the limit equals 1/L = 1/e^4 = e^(-4).

I hope this helps!