From the given problem:

You have an exponential function $f(x)$ with the following end behavior:

$\lim_{x \to -\infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to \infty} f(x) = 0$

Analysis:

1. This describes a decreasing exponential function, where as $x \to -\infty$, the function grows without bound ($f(x) \to \infty$), and as $x \to \infty$, the function approaches zero ($f(x) \to 0$).

2. A possible form of $f(x)$ is: $f(x) = ke^{-x}, \quad \text{where } k > 0.$

   In this case:

   • As $x \to \infty$, $e^{-x} \to 0$, so $f(x) \to 0$.
   • As $x \to -\infty$, $e^{-x} \to \infty$, so $f(x) \to \infty$.

3. The graph of this function typically appears as a curve that starts at a high value (approaching $\infty$) as $x \to -\infty$, and decreases smoothly to approach 0 as $x \to \infty$.

Let me know if you'd like me to explain further or help with specific graph options. Here are some questions to consider:

1. What are the typical shapes of exponential functions based on their growth or decay properties?
2. How does the base of the exponential function affect the steepness of the graph?
3. What are the real-world applications of decreasing exponential functions?
4. How do horizontal asymptotes feature in exponential graphs?
5. Can we relate this to any specific transformations (like reflections or translations)?

Tip: To identify end behavior in graphs, always check how the function behaves as $x \to \infty$ and $x \to -\infty$.