Nice application of dominated convergence theorem

Question

Let $\delta \in \mathbb{R}$, $$f(x)=\frac{sin(x^2)}{x}+\frac{\delta x}{1+x}.$$

Show that $$\operatorname{lim_{n\to \infty}} \int_{0}^{a}f(nx)=a\delta$$ for each $a>0.$ I am unable to find integrable bound for the sequence of function $\{f(nx)\}$. And want to apply Dominated convergence theorem.

Answer 1

$|f(t)|\le t+\delta t\le (1+\delta)a$ if $0 \le t \le a$ and $|f(t)|\le \frac 1 x+\delta\leq (\frac 1 a+\delta)$ if $t >a$. So $|f(t)|\le \max \{(1+\delta)a, (\frac 1 a+\delta)\}$ for all $t$. The constant function $g(t)=\max \{(1+\delta)a, (\frac 1 a+\delta)\}$ is a dominating integrable function for $f(nx)$ on $[0,a]$.