find all asymptotes for f(x) = x
2
(e
1/x − 1)

To find all the asymptotes of the function \( f(x) = x^2 \left( e^{\frac{1}{x}} - 1 \right) \), we need to examine the behavior of the function as \( x \) approaches specific values, usually \( x \to \infty \), \( x \to -\infty \), and near any points where the function might become undefined (e.g., vertical asymptotes).

### Step 1: Investigate as \( x \to \infty \)
As \( x \to \infty \), we have:

\[
\frac{1}{x} \to 0
\]

Thus, \( e^{\frac{1}{x}} \to e^0 = 1 \), and so:

\[
f(x) = x^2 (e^{\frac{1}{x}} - 1) \to x^2 (1 - 1) = 0
\]

Therefore, the function tends to 0 as \( x \to \infty \), indicating that the **horizontal asymptote** is:

\[
y = 0 \quad \text{(as \( x \to \infty \))}
\]

### Step 2: Investigate as \( x \to -\infty \)
As \( x \to -\infty \), similarly, \( \frac{1}{x} \to 0 \) and \( e^{\frac{1}{x}} \to e^0 = 1 \). So:

\[
f(x) = x^2 (e^{\frac{1}{x}} - 1) \to x^2 (1 - 1) = 0
\]

Thus, there is also a horizontal asymptote at:

\[
y = 0 \quad \text{(as \( x \to -\infty \))}
\]

### Step 3: Investigate as \( x \to 0^+ \) (from the right side of 0)
As \( x \to 0^+ \), \( \frac{1}{x} \to \infty \), so \( e^{\frac{1}{x}} \to \infty \). Thus:

\[
f(x) = x^2 \left( e^{\frac{1}{x}} - 1 \right) \sim x^2 \cdot e^{\frac{1}{x}}
\]

Since \( e^{\frac{1}{x}} \to \infty \) much faster than \( x^2 \) decreases, \( f(x) \to \infty \) as \( x \to 0^+ \). Therefore, there is no vertical asymptote as \( x \to 0^+ \), but the function diverges to infinity.

### Step 4: Investigate as \( x \to 0^- \) (from the left side of 0)
As \( x \to 0^- \), \( \frac{1}{x} \to -\infty \), so \( e^{\frac{1}{x}} \to 0 \). Therefore:

\[
f(x) = x^2 \left( e^{\frac{1}{x}} - 1 \right) \sim x^2 \cdot (-1) = -x^2
\]

Thus, as \( x \to 0^- \), \( f(x) \to 0 \).

### Conclusion:
- **Horizontal asymptote**: \( y = 0 \) (both as \( x \to \infty \) and \( x \to -\infty \)).
- **No vertical asymptote** at \( x = 0 \), but the function diverges as \( x \to 0^+ \).

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you find the derivative of \( f(x) = x^2(e^{1/x} - 1) \)?
2. What are the critical points of the function?
3. Can you determine the concavity of the function at different points?
4. What happens to the function when \( x \) approaches zero from both sides?
5. Can you explore limits of other functions with \( e^{1/x} \) terms?

**Tip**: Always check both sides of a function near potential vertical asymptotes for more precise behavior.

find all asymptotes for f(x) = x^2(e^{1/x} − 1)

Explore how to find the horizontal asymptotes of the function f(x) = x^2(e^{1/x} − 1) by analyzing its limits as x approaches infinity, negative infinity, and near zero. Learn key concepts like exponential growth and asymptote behavior, suitable for advanced high school or early college math students.

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