The problem requires evaluating the two-variable limit:

$\lim_{(x, y) \to (0, 0)} \frac{e^{-\frac{1}{x^2 + y^2}}}{\sqrt{x^2 + y^2}}.$

Let's analyze this in detail.

Step 1: Substituting $(x, y) = (0, 0)$ directly

If we substitute $(x, y) = (0, 0)$ directly:

• $x^2 + y^2 = 0$, so the denominator $\sqrt{x^2 + y^2}$ becomes $0$, making the fraction undefined.
• The numerator $e^{-\frac{1}{x^2 + y^2}}$ is also undefined as $x^2 + y^2 = 0$ leads to $e^{-\infty}$.

Thus, direct substitution fails. We proceed with another approach.

Step 2: Converting to polar coordinates

To better handle the behavior near the origin, we convert to polar coordinates: $x = r \cos\theta, \quad y = r \sin\theta, \quad x^2 + y^2 = r^2, \quad \sqrt{x^2 + y^2} = r.$

The limit becomes: $\lim_{(x, y) \to (0, 0)} \frac{e^{-\frac{1}{x^2 + y^2}}}{\sqrt{x^2 + y^2}} = \lim_{r \to 0^+} \frac{e^{-\frac{1}{r^2}}}{r}.$

Step 3: Behavior of the numerator and denominator

1. As $r \to 0^+$:
   • The numerator $e^{-\frac{1}{r^2}} \to e^{-\infty} = 0$.
   • The denominator $r \to 0^+$.

Thus, the fraction $\frac{e^{-\frac{1}{r^2}}}{r}$ has the indeterminate form $\frac{0}{0}$.

Step 4: Dominance of the exponential decay

To resolve the indeterminacy, observe the growth rates:

• $e^{-\frac{1}{r^2}}$ decays extremely rapidly as $r \to 0^+$.
• $r$ tends to $0$ much slower than the decay of $e^{-\frac{1}{r^2}}$.

Since the exponential term dominates, the fraction tends to $0$.

Conclusion

$\lim_{(x, y) \to (0, 0)} \frac{e^{-\frac{1}{x^2 + y^2}}}{\sqrt{x^2 + y^2}} = 0.$

Would you like a detailed breakdown of why the exponential dominates over polynomial terms?

5 Related Questions

1. How does polar coordinate substitution help simplify limits?
2. Why does $e^{-\frac{1}{r^2}}$ decay faster than polynomial terms?
3. What are other techniques for evaluating multivariable limits?
4. Can the limit depend on the path of approach (e.g., along $x$- or $y$-axis)?
5. How do indeterminate forms arise in limits involving exponential functions?

Tip

For multivariable limits, consider symmetry or substitutions like polar coordinates to simplify analysis near the origin.