Evaluate the integral of e^(-x) dx from 0 to infinity

Problem: Evaluate the improper integral $\int_{0}^{\infty} e^{-x} \, dx$.

Steps

1. Rewrite as a limit
   $$\lim_{t \to \infty} \int_{0}^{t} e^{-x} \, dx$$
2. Find the antiderivative
   $$\lim_{t \to \infty} \left[ -e^{-x} \right]_{0}^{t}$$
3. Apply the Fundamental Theorem of Calculus
   $$\lim_{t \to \infty} \left( -e^{-t} - (-e^{0}) \right)$$
4. Simplify the expression
   $$\lim_{t \to \infty} \left( 1 - \frac{1}{e^{t}} \right)$$
5. Evaluate the limit
   $$1 - 0$$