As $\displaystyle v \, \rightarrow \, c$ then $\displaystyle 1-\frac{v^2}{c^2} \, \rightarrow \, 1-1 = 0$
As the denominator of a fraction tends to 0 the variable being measured gets infinitely large, in this case m gets infinitely large.
Since division by 0 is undetermined then an object travelling at c would have undetermined (but very large) mass. This means that nothing can travel at c.
To summarise as $\displaystyle v \, \rightarrow \, c \: \: \text{ , } \: m \, \rightarrow \, \infty$