[-/1 Points] DETAILS MY NOTES SCALCET9 2.6.034. Find the limit. (If the limit is infinite, enter ' \( \infty \) ' or ' \( -\infty \) ', as appropriate. If the limit does not otherwise exist, enter DNE.) \[ \lim _{x \rightarrow \infty}\left(e^{-x}+9 \cos (5 x)\right) \]

We need to evaluate the limit as x approaches infinity for the expression e^(–x) + 9 cos(5x). Step 1: Compute the limit of e^(–x) as x approaches infinity. Since e^(–x) → 0 when x → ∞, we have: lim (x→∞) e^(–x) = 0. Step 2: Consider the term 9 cos(5x). The cosine function oscillates between –1 and 1 for all x, so cos(5x) oscillates between –1 and 1 regardless of x. When multiplied by 9, 9 cos(5x) oscillates between –9 and 9. Since the oscillatory behavior persists and does not settle to a single value, the overall limit does not exist. Final answer: DNE.