Limit (eˣ−1−x) over x²

Problem

Compute $\lim_{x\to 0}\dfrac{e^x - 1 - x}{x^2}$ using Taylor.

Solution

From the expansion $e^x = 1 + x + \dfrac{x^2}{2} + o(x^2)$, the numerator becomes: $e^x - 1 - x = \frac{x^2}{2} + o(x^2).$ Hence: $\frac{e^x - 1 - x}{x^2} = \frac{x^2/2 + o(x^2)}{x^2} = \frac{1}{2} + o(1) \xrightarrow{x\to 0} \boxed{\frac{1}{2}}.$

Links

Topics: Derivatives
Concepts: Limit · Standard expansion
Methods: Limits via Taylor
Skills: Compute limits
Exercise type: Limit computation