Evaluate the following limits :
`lim_(x to infty)e^(-x)`

Answer

To evaluate the limit \( \lim_{x \to \infty} e^{-x} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \( e^{-x} \). We can rewrite this using the property of exponents:
\[ e^{-x} = \frac{1}{e^x} \]

### Step 2: Analyze the limit
Now we need to evaluate the limit:
\[ \lim_{x \to \infty} e^{-x} = \lim_{x \to \infty} \frac{1}{e^x} \]

### Step 3: Determine the behavior of \( e^x \)
As \( x \) approaches infinity, \( e^x \) also approaches infinity. Therefore, we have:
\[ \lim_{x \to \infty} e^x = \infty \]

### Step 4: Evaluate the limit of the fraction
Now substituting this back into our limit, we get:
\[ \lim_{x \to \infty} \frac{1}{e^x} = \frac{1}{\infty} \]

### Step 5: Conclude the limit
Since \( \frac{1}{\infty} \) approaches 0, we conclude that:
\[ \lim_{x \to \infty} e^{-x} = 0 \]

### Final Answer
Thus, the limit is:
\[ \boxed{0} \]