Multivariable limits

Recently, I've just learn about multivariate limits in my Advanced Calculus class. The concept is such that there are infinite possible approaches in solving limit problems (see picture). Why would it not be sufficient to just consider the direct displacement of our approach? If there is a direct displacement from all directions, it would form an area such that all paths taken falls within the area. Hence, if the limit of the direct displacement exists, we can safely conclude that the limits of all paths exists.

If that is the case, we can simply let y=mx,mRy=mx, m\in\mathbb{R} and solve the limits. I will show my approach in multivariate problems in the following forms. Consider 44 cases, lim(x,y)(0,0)f(x,y)\lim_{(x, y)\rightarrow(0, 0)}f(x, y), lim(x,y)(a,0)f(x,y)\lim_{(x, y)\rightarrow(a, 0)}f(x, y), lim(x,y)(0,b)f(x,y)\lim_{(x, y)\rightarrow(0, b)}f(x, y) and lim(x,y)(a,b)f(x,y)\lim_{(x, y)\rightarrow(a, b)}f(x, y) where a,b0a, b\neq0. In all cases, let y=mxy=mx.

Case 11: lim(x,y)(0,0)f(x,y)\lim_{(x, y)\rightarrow(0, 0)}f(x, y) x,y0,m=R\because x, y\rightarrow0, \therefore m=\mathbb{R} lim(x,y)(0,0)f(x,y)=lim(x,mx)(0,0)f(x,mx)lim(x,y)(0,0)f(x,y)=limx0f(x,mx)\therefore\lim_{(x, y)\rightarrow(0, 0)}f(x, y)=\lim_{(x, mx)\rightarrow(0, 0)}f(x, mx)\Rightarrow \lim_{(x, y)\rightarrow(0, 0)}f(x, y)=\lim_{x\rightarrow0}f(x, mx)

If the multivariate limit exists, then lim(x,y)(0,0)f(x,y)=limx0)f(x,mx)=L,LR\lim_{(x, y)\rightarrow(0, 0)}f(x, y)=\lim_{x\rightarrow0)}f(x, mx)=L, L\in\mathbb{R}, Else, limx0f(x,mx)=g(m)\lim_{x\rightarrow0}f(x, mx)=g(m) and the limit does not exist.

Case 22: lim(x,y)(a,0)f(x,y)\lim_{(x, y)\rightarrow(a, 0)}f(x, y) xa,y0,m0\because x\rightarrow a, y\rightarrow0, \therefore m\rightarrow0 lim(x,y)(a,0)f(x,y)=lim(x,m)(0,0)f(x,mx)Case 1\therefore\lim_{(x, y)\rightarrow(a, 0)}f(x, y)=\lim_{(x, m)\rightarrow(0, 0)}f(x, mx)\Rightarrow\text{Case 1} And repeat the process of case 11 to obtain the answer.

Case 33: lim(x,y)(0,b)f(x,y)\lim_{(x, y)\rightarrow(0, b)}f(x, y) x0,yb,mb0\because x\rightarrow0, y\rightarrow b, \therefore m\rightarrow\frac{b}{0}\rightarrow\infty lim(x,y)(0,b)f(x,y)=lim(x,m)(0,)f(x,mx)\therefore\lim_{(x, y)\rightarrow(0, b)}f(x, y)=\lim_{(x, m)\rightarrow(0, \infty)}f(x, mx) Then, let m=1tm=\frac{1}{t}. Hence, t0t\rightarrow0 as mm\rightarrow\infty. This gives us lim(x,m)(0,)f(x,mx)=lim(x,1t)(0,0)f(x,xt)Case 1\lim_{(x, m)\rightarrow(0, \infty)}f(x, mx)=\lim_{(x, \frac{1}{t})\rightarrow(0, 0)}f(x, \frac{x}{t})\Rightarrow\text{Case 1}

Case 44: lim(x,y)(a,b)f(x,y)\lim_{(x, y)\rightarrow(a, b)}f(x, y) xa,yb,m=ba\because x\rightarrow a, y\rightarrow b, \therefore m=\frac{b}{a} lim(x,y)(a,b)f(x,y)=limxaf(x,bax)\therefore\lim_{(x, y)\rightarrow(a, b)}f(x, y)=\lim_{x\rightarrow a}f(x, \frac{b}{a}x)

Note: Reducing the multivariate limits to a single variable limit allows the use of L'hopital's Rule. I have tested on many (but not all) problems to prove that the method works.

Any faults about the idea?

Note by Shaun Loong
5 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...