Find the limit as x approaches 0 of (e^x - 1 - x)/x^2

Problem: Find the limit $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$.

Steps

1. Identify indeterminate form
   $$\frac{e^0 - 1 - 0}{0^2} = \frac{0}{0}$$
2. Apply L'Hopital's Rule
   $$\lim_{x \to 0} \frac{\frac{d}{dx}(e^x - 1 - x)}{\frac{d}{dx}(x^2)} = \lim_{x \to 0} \frac{e^x - 1}{2x}$$
3. Identify indeterminate form again
   $$\frac{e^0 - 1}{2(0)} = \frac{0}{0}$$
4. Apply L'Hopital's Rule a second time
   $$\lim_{x \to 0} \frac{\frac{d}{dx}(e^x - 1)}{\frac{d}{dx}(2x)} = \lim_{x \to 0} \frac{e^x}{2}$$
5. Evaluate the limit by direct substitution
   $$\frac{e^0}{2} = \frac{1}{2}$$