Find the Limit of a function

Question

Find the limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)

lim x → ∞ (e^−x + 4 cos(3x))

Answer 1

We have lim x → ∞ e^−x = 0 and lim x → ∞ 4 cos(3x) DNE. Hence lim x → ∞ (e^−x + 4 cos(3x)) DNE.

Remark: If we change it a little bit, lim x → ∞ (e^x + 4 cos(3x)) = ∞. Because e^x → ∞ and 4 cos(3x) is bounded between -4 and 4 as x→∞.