Lots of Limits

For the 1st one .... approaching $(0,0)$ through $y=mx$ .

we get the limit equals $\displaystyle \frac{m}{1+m^2}$ which is different for different values of m .

Hence limit Does not exist.

for 2:-

approaching $(0,0)$ through $\displaystyle y=x^2$ , we see that the right hand limit tends to positive $\infty$ and left hand limit tends to negative $\infty$ . Hence limit does not exist.

for 3 :-

putting $\displaystyle y = mx^3$ we see that the limit equals $\displaystyle \frac{m}{1+m^3}$ which is different for different values of m . Hence limit does not exist.

for 4:-

we see that the limit equals $\displaystyle \left(\lim_{y\to 1}\frac{1}{y+1}\right)\left(\lim_{x\to 0}\left(\frac{\tan(x)}{x}\right)^{2}\right)$ .

which equals $\displaystyle \frac{1}{2}$ .

Hence $D$ is the only correct option