如何计算多元极限？含特定三元无穷极限求解问询

Hey there! Let's walk through how to handle limits for multivariable functions as the variables head to infinity, then solve that triple limit you're asking about.

When dealing with limits like $\lim_{|(x_1,x_2,...,x_n)| \to +\infty} f(x_1,x_2,...,x_n)$, here are key approaches to keep in mind:

• Understand the definition: $|(x_1,...,x_n)| \to +\infty$ means the Euclidean norm $\sqrt{x_1^2 + x_2^2 + ... + x_n^2}$ goes to infinity. Unlike single-variable limits, we need to check that the function behaves consistently no matter which path we take to infinity (if different paths give different results, the limit doesn't exist).
• Bounding/Comparison: Use inequalities to estimate the function's behavior—show it stays above (or below) some expression that clearly tends to a specific value (like $+\infty$ or $-\infty$).
• Coordinate transformations: For 2D or 3D cases, switch to polar/spherical coordinates to turn the problem into a single-variable limit as the radius $r \to +\infty$, while ensuring the result doesn't depend on angles (since angles represent different paths).
• Dominant term analysis: Identify the term in the function that grows the fastest. Lower-order terms become negligible as variables go to infinity, so we can focus on the dominant term to find the limit (but always verify this with a rigorous proof).

Let's break down this expression and prove its limit step by step:

First, rewrite the expression to highlight individual terms:
$$x^4 + (y^2 + 3y) + (z^2 - z) - x$$

Step 1: Analyze each term's growth rate #

• $x^4$ is a fourth-order infinity—it grows far faster than any lower-order terms (like $x$, $y^2$, or $z^2$) as $|x|$ increases.
• $y^2 + 3y$ can be rewritten by completing the square: $(y + \frac{3}{2})^2 - \frac{9}{4}$. As $|y| \to +\infty$, this tends to $+\infty$ (second-order infinity).
• $z^2 - z$ similarly becomes $(z - \frac{1}{2})^2 - \frac{1}{4}$, which also tends to $+\infty$ as $|z| \to +\infty$.
• The $-x$ term is a first-order infinity, which is negligible compared to $x^4$ when $|x|$ is large.

Step 2: Rigorous proof via bounding #

For any arbitrarily large positive number $M$, we need to find a radius $R > 0$ such that if $\sqrt{x^2 + y^2 + z^2} > R$, then the expression is greater than $M$.

Start with some inequalities:

• When $|x| \geq 1$, $x^4 \geq x^2$. So the original expression is at least:
  $$x^2 + y^2 + z^2 - |x| - 3|y| - |z|$$
• Let $r = \sqrt{x^2 + y^2 + z^2}$. We know $|x| \leq r$, $|y| \leq r$, $|z| \leq r$. Substitute these into the inequality:
  $$r^2 - |x| - 3|y| - |z| \geq r^2 - r - 3r - r = r^2 - 5r$$

Now solve for $r$ such that $r^2 - 5r > M$. The quadratic equation $r^2 -5r - M =0$ has roots $r = \frac{5 \pm \sqrt{25 + 4M}}{2}$. We take the positive root: $r > \frac{5 + \sqrt{25 +4M}}{2}$.

If we set $R$ to be the maximum of this root and 1 (to ensure $|x| \geq1$), then whenever $|(x,y,z)| > R$, the original expression is greater than $M$. This proves the limit tends to $+\infty$.

Step 3: Path verification (to confirm consistency) #

We can check a few paths to make sure all lead to the same result:

• Path 1: $y=0, z=0, x \to +\infty$ → expression becomes $x^4 -x \to +\infty$
• Path 2: $x=0, z=0, y \to +\infty$ → expression becomes $y^2 +3y \to +\infty$
• Path 3: $x=t, y=t, z=t, t \to +\infty$ → expression becomes $t^4 + t^2 + t^2 -t +3t -t = t^4 +2t^2 +t \to +\infty$

All paths confirm the expression tends to $+\infty$.

内容的提问来源于stack exchange，提问作者Asim90