Answer 1

\[\lim_{x \to 0} \left[ \frac{e^{3 + x} - \sin x - e^3}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \left( \frac{e^{3 + x} - e^3}{x} \right) - \frac{\sin x}{x} \right]\]
\[ = \lim_{x \to 0} \left[ e^3 \left( \frac{e^x - 1}{x} \right) - \frac{\sin x}{x} \right]\]
\[ = e^3 \times 1 - 1\]
\[ = e^3 - 1\]