In the context of a measure theory and Lebesgue integration course I have to compute the limit $$\lim_{n\to \infty} \int_{1}^{\infty} \exp(-x)(\text{cos}(x))^n dx $$
To me this looks like the application of some convergence theorem, either the dominated or monotone one. I first checked some general properties relating to the convergence theorems:
Pointwise convergence: Since $\vert \text{cos}(x) \vert \leq 1$, it holds that $\vert (\text{cos}(x))^n \vert \leq 1 \forall n \in \mathbb{N}$. Hence I know that for the pointwise limit holds that $$-\exp(-x) \leq \lim_{n \to \infty} \exp(-x) (\text{cos}(x))^n \leq \exp(-x)$$ But what can I infer about pointwise convergence from this? Since cosine oscillates between 1 and -1 there seems to be no pointwise limit. But there has to be in order to apply either one of the convergence theorems. So what do I miss here?
Dominating function: Let's apply the dominated convergence theorem. For this we need a non-negative function $g \in L(\lambda)$ for which it holds that $$\vert f_n(x) \vert \leq g(x) \quad \forall x \in [1,\infty),\quad \forall n \in \mathbb{N}$$. Obviously, a good candidate is $g(x) = \exp(-x)$ since it is non-negative and $$\int_{[1,\infty)} \vert\exp(-x)\vert d\lambda(x) < \infty$$
Now we can apply the dominated convergence theorem and write $$ \lim_{n \to \infty} \int_{0}^{\infty} \exp(-x) (\text{cos}(x))^n = \int_{0}^{\infty} \lim_{n \to \infty} \exp(-x) (\text{cos}(x))^n d\lambda(x)$$
Now all I need to do is evaluate the actual limit $$ \lim_{n \to \infty} \exp(-x) (\text{cos}(x))^n$$ but I don't see it since $(\text{cos(x)})^n$ jumps back and forth for $\lim_{n \to \infty}$.
Your ansatz with the dominated convergence theorem is right. A last step is missing: We know that for $f_n(x)=\exp(-x)(\text{cos}(x))^n$ we have $f_n(x) \to \exp(-x) \cdot 0$ for all $x \in \mathbb{R} \setminus \pi\mathbb{Z}$. But $\pi\mathbb{Z}$ is a set of measure zero in $\mathbb{R}$ (since it is discrete). Hence $f_n(x) \to 0$ almost everywhere. It follows
$$\lim_{n \to \infty} \int_{0}^{\infty} \exp(-x) (\text{cos}(x))^n = \int_{0}^{\infty} \lim_{n \to \infty} \exp(-x) (\text{cos}(x))^n d\lambda(x) $$ $$= \int_{0}^{\infty} 0 d\lambda(x) = 0$$