Math Problem Statement
solve limit when x goes minus infinite and pus infinite and function is x*e^(-2x)
Solution
We are tasked with finding the limits of the function as approaches both negative infinity () and positive infinity ().
1. Limit as
First, let's consider the limit of as .
As :
- because the exponential function decays very rapidly for large positive .
- .
So, we have an indeterminate form . To resolve this, let's rewrite the expression by recognizing that can be written as :
Now, we evaluate:
This is a case of the limit , where the exponential function grows much faster than the polynomial. Therefore, this limit is:
2. Limit as
Now, let's consider the limit of as .
As :
- because the exponential grows very rapidly for large negative .
- .
Thus, the function becomes , which suggests that the limit will be negative infinity:
Summary of Limits:
Let me know if you'd like further details or have any questions!
Here are 5 follow-up questions:
- How does the behavior of exponential functions influence the limits of functions involving products?
- What other methods can we use to solve indeterminate forms like ?
- Can L'Hopital's Rule be applied to this limit problem? If so, how?
- What happens to when is a positive constant other than 2?
- How does the behavior of change if is negative?
Tip: For limits involving both polynomials and exponentials, exponentials generally dominate as .
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Math Problem Analysis
Mathematical Concepts
Calculus
Limits at Infinity
Exponential Functions
Formulas
lim (x → +∞) [x * e^(-2x)] = 0
lim (x → -∞) [x * e^(-2x)] = -∞
Theorems
Indeterminate Forms
Polynomial vs Exponential Growth
L'Hôpital's Rule (Optional for solving indeterminate forms)
Suitable Grade Level
College Level (Calculus I/II)