1. **State the problem:** Find the limit of the function $f(x) = e^{\frac{1}{x+2}}$ as $x$ approaches 2. 2. **Recall the limit definition and properties:** The exponential function $e^y$ is continuous for all real $y$, so we can find the limit by evaluating the exponent's limit first. 3. **Evaluate the inner limit:** $$\lim_{x \to 2} \frac{1}{x+2} = \frac{1}{2+2} = \frac{1}{4}$$ 4. **Apply the continuity of the exponential function:** $$\lim_{x \to 2} e^{\frac{1}{x+2}} = e^{\lim_{x \to 2} \frac{1}{x+2}} = e^{\frac{1}{4}}$$ 5. **Final answer:** $$\boxed{e^{\frac{1}{4}}}$$ This means as $x$ gets closer to 2, the value of $e^{\frac{1}{x+2}}$ approaches $e^{\frac{1}{4}}$.