Limits and Continuity · Medium problem

The rational function $f(x)=\dfrac{x^2+kx-6}{x-2}$ has an infinite discontinuity at $x=2$ for every value of $k$ except one, where instead $f$ has a removable discontinuity. Find that value of $k$, plus $\displaystyle\lim_{x\to2}f(x)$ for that $k$.