Q.1. $\lim_{x\rightarrow 0}$ xlogx = $\lim_{x\rightarrow 0}$ $\frac{log x}{\frac{1}{x}}$ $\frac{infinity}{infinity}$ form. Use L'hospital rule
= $\lim_{x\rightarrow 0}$ $\frac{\frac{1}{x}}{\frac{-1}{x^{2}}}$
= $\lim_{x\rightarrow 0}$ -x
= 0
Q.2. $\lim_{x\rightarrow Inf}$ $\frac{sinx}{x}$ = 0
Here, sin x is a bounded function i.e. its range is fixed [-1,1] whatever may be the value of x.
So, $\lim_{x\rightarrow Inf}$ $\frac{sinx}{x}$ will give a finite value divided by an infinite value which is 0.