Taylor series about a point

Problem 7.405 · hard

1. \[ \left[\begin{matrix}\left. e^{x} \right|_{\substack{ x=2 }}\\\left. \frac{d}{d x} e^{x} \right|_{\substack{ x=2 }}\\\left. \frac{d^{2}}{d x^{2}} e^{x} \right|_{\substack{ x=2 }}\\\left. \frac{d^{3}}{d x^{3}} e^{x} \right|_{\substack{ x=2 }}\end{matrix}\right] = \left[\begin{matrix}e^{2}\\e^{2}\\e^{2}\\e^{2}\end{matrix}\right] \]
   f and its first three derivatives at a.✓ Proved
2. T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
   Reviewed
3. \[ \lim_{x \to 2^+}\left(\frac{- \frac{\left(x - 2\right)^{3} e^{2}}{6} - \frac{\left(x - 2\right)^{2} e^{2}}{2} - \left(x - 2\right) e^{2} + e^{x} - e^{2}}{\left(x - 2\right)^{3}}\right) = 0 \]
   T agrees with f to third order at a.✓ Proved