País Vasco 2012-P5

Calcular \(\require{AMSmath}\)$$\lim_{n\to\infty} \begin{vmatrix} ~1+x & -x & 0 & 0 & \dots & 0 & 0 \\[0.3em] -\dfrac{1}{2} & 1+\dfrac{x}{2} & -x & 0 & \dots & 0 & 0 \\[0.3em] 0 & -\dfrac{1}{3} & 1+\dfrac{x}{3} & -x & \dots & 0 & 0 \\[0.3em] 0 & 0 & -\dfrac{1}{4} & 1+\dfrac{x}{4} & \dots & 0 & 0 \\[0.3em] \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\[0.3em] 0 & 0 & 0 & 0 & \dots & 1+\dfrac{x}{n-1} & -x \\[0.3em] 0 & 0 & 0 & 0 & \dots & -\dfrac{1}{n} & 1+\dfrac{x}{n}~ \end{vmatrix}$$