$\displaystyle \lim_{x\to\\1^+}\frac{x+1}{x\sin{{\pi}x}}$
I was never any good at these. >.<
I didn't notice the "right hand limit" part.
There are several ways to rearrange the expression, including
$\displaystyle \frac{x + 1}{x \sin{\pi x}} = \frac{x}{x\sin{\pi x}} + \frac{1}{x\sin{\pi x}}$
$\displaystyle = \frac{1}{\sin{\pi x}} + \frac{1}{x \sin{\pi x}}$
$\displaystyle = \frac{1}{\sin{\pi x}}\left(1 + \frac{1}{x}\right)$.
All will lead to the same conclusion though.
You most definitely can NOT use L'Hospital's Rule in this case.
To use L'Hospital, the expression must tend to an indefinite form, i.e. $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$.
But like I showed earlier, the expression tends to $\displaystyle \frac{2}{0}$.