We are tasked with finding the limits of the function $f(x) = x \cdot e^{-2x}$ as $x$ approaches both negative infinity ($x \to -\infty$) and positive infinity ($x \to +\infty$).

1. Limit as $x \to +\infty$

First, let's consider the limit of $f(x) = x \cdot e^{-2x}$ as $x \to +\infty$.

As $x \to +\infty$:

• $e^{-2x} \to 0$ because the exponential function decays very rapidly for large positive $x$.
• $x \to +\infty$.

So, we have an indeterminate form $\infty \cdot 0$. To resolve this, let's rewrite the expression by recognizing that $e^{-2x}$ can be written as $\frac{1}{e^{2x}}$:

$f(x) = x \cdot e^{-2x} = \frac{x}{e^{2x}}.$

Now, we evaluate:

$\lim_{x \to +\infty} \frac{x}{e^{2x}}.$

This is a case of the limit $\frac{\text{polynomial}}{\text{exponential}}$, where the exponential function grows much faster than the polynomial. Therefore, this limit is:

$\lim_{x \to +\infty} \frac{x}{e^{2x}} = 0.$

2. Limit as $x \to -\infty$

Now, let's consider the limit of $f(x) = x \cdot e^{-2x}$ as $x \to -\infty$.

As $x \to -\infty$:

• $e^{-2x} = e^{2|x|} \to \infty$ because the exponential grows very rapidly for large negative $x$.
• $x \to -\infty$.

Thus, the function becomes $(-\infty) \times \infty$, which suggests that the limit will be negative infinity:

$\lim_{x \to -\infty} x \cdot e^{-2x} = -\infty.$

Summary of Limits:

$\lim_{x \to +\infty} x \cdot e^{-2x} = 0,$ $\lim_{x \to -\infty} x \cdot e^{-2x} = -\infty.$

Let me know if you'd like further details or have any questions!

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Here are 5 follow-up questions:

1. How does the behavior of exponential functions influence the limits of functions involving products?
2. What other methods can we use to solve indeterminate forms like $\infty \cdot 0$?
3. Can L'Hopital's Rule be applied to this limit problem? If so, how?
4. What happens to $f(x) = x \cdot e^{-kx}$ when $k$ is a positive constant other than 2?
5. How does the behavior of $f(x) = x \cdot e^{kx}$ change if $k$ is negative?

Tip: For limits involving both polynomials and exponentials, exponentials generally dominate as $x \to +\infty$.