Math Problem Statement

Soit f la fonction définie sur ℝ par : f(x) = 7e^{-x^4}. Le but de cet exercice est de calculer par étapes la limite de f en +∞. 1. La fonction f est de la forme ke^u avec : u(x) = ? et k = ?

Solution

The problem asks us to analyze the function f(x)=7ex4f(x) = 7e^{-x^4}, write it in the form keu(x)k e^{u(x)}, and later compute the limit of f(x)f(x) as x+x \to +\infty.

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

We can rewrite f(x)=7ex4f(x) = 7e^{-x^4} as: f(x)=keu(x),f(x) = k e^{u(x)}, where:

  • k=7k = 7,
  • u(x)=x4u(x) = -x^4.

Thus: u(x)=x4andk=7.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)=x4u(x) = -x^4 as x+x \to +\infty?
  2. Why does the exponential term ex4e^{-x^4} approach 0 as x+x \to +\infty?
  3. How would you compute limxf(x)\lim_{x \to -\infty} f(x) for the same function?
  4. Can you find the derivative f(x)f'(x) to determine how the function behaves?
  5. What is the effect of the constant kk 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 00, \infty, or remains finite.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Limits
Asymptotic Behavior

Formulas

f(x) = ke^{u(x)}

Theorems

Exponential Growth and Decay Behavior

Suitable Grade Level

Grades 10-12