I prove to limit exist, but i can't find this limit
$a_n=\lim \limits_{n\to\infty}\prod \limits_{j=1}^n \left[ 1+\frac{1}{j} \left(\cos \left(\frac{tj}{n} \right)-1 \right) \right]$
$\to \ln a_n=\lim \limits_{n\to\infty}\sum_{j=1}^n \ln\left[ 1+\frac{1}{j} \left(\cos \left(\frac{tj}{n} \right)-1 \right) \right]$
Using Taylor's expansion: $\ln(1+x)=x+O(x)$
Hence: $ \ln\left[ 1+\frac{1}{j} \left(\cos \left(\frac{tj}{n} \right)-1 \right) \right]=\frac{1}{j} \left(\cos \left(\frac{tj}{n} \right)-1 \right)+O(\cdots)$
$\to \ln a_n=\lim \limits_{n\to\infty}\sum_{j=1}^n \left[ \frac{1}{j} \left(\cos \left(\frac{tj}{n} \right)-1 \right) +O(\cdots)\right] =\int_{0}^{1}\frac{\cos(tx)-1}{x}dx$
Because: $\int_{0} ^{1} \left(\cos(tx)-1\right) dx\leq 3$
Hence, integral $\int_{0}^{1}\frac{\cos(tx)-1}{x}dx$ is convergent integral
Wolfram alpha