$a)\underset{x\to \infty }{\lim }\left(100+x-x^3\right)=\underset{x\to \infty }{\lim }{x}^3\left(\frac{100}{x^3}+\frac{1}{x^2}-1\right)=\left[\infty \cdot \left(0+0-1\right)\right]=-\infty$  

 

$b)\underset{x\to -\infty }{\lim }{\left(x^3+x\right)}^3=\underset{x\to -\infty }{\lim }{\left[x^3\left(1+\frac{1}{x^2}\right)\right]}^3=\underset{x\to -\infty }{\lim }{x}^9{\left(1+\frac{1}{x^2}\right)}^3=\left[-\infty \cdot {\left(1+0\right)}^3\right]=-\infty$  

 

$c)\underset{x\to -\infty }{\lim }\left(\frac{x^4}{10}+2x^3+x^2\right)=\underset{x\to -\infty }{\lim }{x}^4\left(\frac{1}{10}+\frac{2}{x}+\frac{1}{x^2}\right)=\left[\infty \cdot \left(\frac{1}{10}-0+0\right)\right]=\infty \cdot \frac{1}{10}=\infty$