1. **Stating the problem:** Find the limit \( \lim_{x \to \infty} \left(5 + \frac{1}{x}\right) \). 2. **Formula and rules:** The limit of a sum is the sum of the limits, provided each limit exists. That is, \( \lim_{x \to \infty} (f(x) + g(x)) = \lim_{x \to \infty} f(x) + \lim_{x \to \infty} g(x) \). 3. **Apply the limit:** \[ \lim_{x \to \infty} \left(5 + \frac{1}{x}\right) = \lim_{x \to \infty} 5 + \lim_{x \to \infty} \frac{1}{x} \] 4. **Evaluate each limit:** - \( \lim_{x \to \infty} 5 = 5 \) because 5 is a constant. - \( \lim_{x \to \infty} \frac{1}{x} = 0 \) because as \( x \) grows larger, \( \frac{1}{x} \) approaches zero. 5. **Combine results:** \[ 5 + 0 = 5 \] **Final answer:** \[ \lim_{x \to \infty} \left(5 + \frac{1}{x}\right) = 5 \]