Calculating $\lim_{x \to a} \frac{x^2 - (a+1)x + a}{x^3-a^3}$ using L'Hospital I tried to calculate this limit:
$$\lim_{x \to a} \frac{x^2 - (a+1)x + a}{x^3-a^3}$$
Using L'Hospital's rule I get:
$$\lim_{x \to a} \frac{2x - (a+1)}{3x^2} = \frac{2a - (a+1)}{3a^2} = 0$$
But actually the limit is
$$\lim_{x \to a} \frac{2x - (a+1)}{3x^2} = \frac{a-1}{3a^2}$$
Can you help me find my mistake?

$\lim_{x \to a} \frac{2x - (a+1)}{3x^2} = \frac{2a - (a+1)}{3a^2} = 0$ is incorrect.
If you check more carefully, you'll see that $\lim_{x \to a} \frac{2x - (a+1)}{3x^2} = \frac{2a - (a+1)}{3a^2} = \frac{2a-a-1}{3a^2}=\frac{a-1}{3a^2}$
