We are tasked with evaluating the limit:

$\lim_{x \to 0} \left( e^x - \left(1 + x + \frac{x^2}{2} \right) \right)$

Step-by-Step Solution:

1. Recall the Taylor expansion of $e^x$ around $x = 0$:

   $e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots$

2. Subtract the polynomial $1 + x + \frac{x^2}{2}$ from $e^x$:

   $e^x - \left( 1 + x + \frac{x^2}{2} \right) = \left( 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \right) - \left( 1 + x + \frac{x^2}{2} \right)$

   Simplifying this expression:

   $e^x - \left( 1 + x + \frac{x^2}{2} \right) = \frac{x^3}{6} + \frac{x^4}{24} + \cdots$

3. Take the limit as $x \to 0$:

   Since each term in the series starting from $\frac{x^3}{6}$ involves powers of $x$ higher than 2, all these terms vanish as $x \to 0$.

   $\lim_{x \to 0} \left( \frac{x^3}{6} + \frac{x^4}{24} + \cdots \right) = 0$

Thus, the limit is:

$\boxed{0}$

Let me know if you need further details or clarifications!

Related questions:

1. What is the Taylor series of $\sin(x)$ around $x = 0$?
2. How can we apply L'Hopital's rule to limits involving indeterminate forms?
3. What is the general form of a Taylor series?
4. How do we expand $\ln(1+x)$ around $x = 0$?
5. Can you provide an example of a limit using a different function, such as $\cos(x)$?

Tip: Taylor expansions are useful for approximating functions near a specific point, especially when dealing with limits!