Given a set of points $\{(x_k,y_k)\}_{i=0,...,n}$ we can modify Lagrange's interpolation method as follows:
If we define $$L(x) := \prod_{i=0}^{n} (x-x_i)$$
then the Lagrange cardinal polynomial becomes $$\phi_k(x) = \frac{1}{\prod_{i=0, i \neq k}^{n}(x_k-x_i)}*\frac{L(x)}{x-x_k} = \frac{1}{L^{\prime}(x_k)}*\frac{L(x)}{x-x_k}$$
If we define $$\lambda_k := \frac{1}{L^{\prime}(x_k)}$$
then we have $$\phi_k(x) = L(x)*\frac{\lambda_k}{x-x_k}$$
and so the Lagrange interpolation polynomial becomes
$$p_n(x) = L(x) \sum_{k=0}^{n} \frac{\lambda_k}{x-x_k}*y_k$$
My question is: Isn't this function not defined at the points $x=x_k$? The denominator in the summand would be zero for such $x$.