Given ,
$\large \lim_{x->\infty}\frac{2x^{3}+3x-5}{5x^{3}+1}$
dividing both numerator and denominator by $x^{3}$ we get,
=$\large \lim_{x->\infty}\frac{2+\frac{3}{x^{2}}-\frac{5}{x^{3}}}{5+\frac{1}{x^{3}}}$
putting the value of limit ,
=$\large \lim_{x->\infty}\frac{2+0-0}{5+0}$ [as $\infty$ in denominator makes $\large \frac{3}{x^{2}}$,$\large \frac{5}{x^{3}}$,$\large \frac{1}{x^{3}}$ terms 0]
=$\large \frac{2}{5}$