A hard limit with integral sign

Question

$$\displaystyle\underset{m\to +\infty }{\mathop{\lim }}\,\left[ \underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\int_{0}^{x}{{{\left( \underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1-\frac{1}{n} \right)}^{n}} \right)}^{{{t}^{m}}}}t\text{d}t} \right]$$

Answer 1

$$\underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1-\frac{1}{n} \right)}^{n}}=e^{-1}$$ and $$\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\int_{0}^{x}{{e^{-t^m}}t\text{d}t}=0,$$ so the answer is $0$.