The problem asks us to analyze the function $f(x) = 7e^{-x^4}$, write it in the form $k e^{u(x)}$, and later compute the limit of $f(x)$ as $x \to +\infty$.

1. Expressing $f(x)$ in the form $k e^{u(x)}$:

We can rewrite $f(x) = 7e^{-x^4}$ as: $f(x) = k e^{u(x)},$ where:

• $k = 7$,
• $u(x) = -x^4$.

Thus: $u(x) = -x^4 \quad \text{and} \quad k = 7.$

Let me know if you'd like a step-by-step breakdown or have additional questions.

Related Questions to Expand Understanding

1. What happens to $u(x) = -x^4$ as $x \to +\infty$?
2. Why does the exponential term $e^{-x^4}$ approach 0 as $x \to +\infty$?
3. How would you compute $\lim_{x \to -\infty} f(x)$ for the same function?
4. Can you find the derivative $f'(x)$ to determine how the function behaves?
5. What is the effect of the constant $k$ on the overall shape of the function?

Quick Tip

When analyzing limits involving exponential functions, focus on the exponent's growth or decay rate, as this often determines whether the exponential term approaches $0$, $\infty$, or remains finite.