Use limit notation to describe the unbounded behavior of the given function as x approaches −∞.

v(x)=e−4x−7

To describe the unbounded behavior of the function \( v(x) = e^{-4x} - 7 \) as \( x \to -\infty \) using limit notation, we need to analyze the function's behavior in that domain.

1. The term \( e^{-4x} \) is the key component to examine. As \( x \to -\infty \), the exponent \( -4x \to +\infty \), and since \( e^{\infty} \to \infty \), the exponential term \( e^{-4x} \) grows without bound.
   
2. Therefore, as \( x \to -\infty \), the function \( v(x) = e^{-4x} - 7 \) becomes increasingly large because \( e^{-4x} \) dominates and grows rapidly. The subtraction of 7 does not change this trend significantly.

Thus, in limit notation, the behavior of \( v(x) \) as \( x \to -\infty \) can be described as:

\[
\lim_{x \to -\infty} v(x) = +\infty
\]

Would you like more details on this explanation or have any questions?

---

Here are 5 related questions to expand your understanding:

1. What happens to the function \( v(x) = e^{-4x} - 7 \) as \( x \to \infty \)?
2. How does the rate of growth of \( e^{-4x} \) compare to \( e^{x} \) as \( x \to -\infty \)?
3. What is the horizontal asymptote of the function \( v(x) = e^{-4x} - 7 \)?
4. How does the value of \( e^{-4x} \) behave for positive values of \( x \)?
5. How would the function change if the exponent was positive, i.e., \( v(x) = e^{4x} - 7 \)?

**Tip:** When working with exponential functions, always analyze the sign of the exponent to predict how the function behaves as \( x \to \infty \) or \( x \to -\infty \).

Use limit notation to describe the unbounded behavior of the given function as x approaches −∞.

v(x) = e^{-4x} - 7

Explore how to use limit notation to describe the unbounded behavior of the function v(x) = e^{-4x} - 7 as x approaches −∞. Understand key mathematical concepts such as limits, exponential growth, and asymptotic behavior. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation