The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$.
$f(x):=\lim\limits_{n \rightarrow \infty}{f_n(x)}$ is lebesgue integrable.
And $\int_\mathbb{R} f(x) d\lambda(x) \neq \lim\limits_{n \rightarrow \infty}{\int_\mathbb{R} f_n(x) d\lambda(x)}$.
Does this violate the dominated convergence theorem?
[All credit goes to Gautam Shenoy]
$\int_\mathbb{R} f(x) d\lambda(x) \neq \lim\limits_{n \rightarrow \infty}{\int_\mathbb{R} f_n(x) d\lambda(x)}$ does not violate the DCT, because you need a dominating function $g$ with $|f_n|\leq g$.
$f$ does not qualify for this, as it equals $0$ for all $x\in\mathbb{R}$