Multivariable Limits: The Two-Path Trap (and the 4-Step Method That Always Works)

Let’s tackle multivariable limits together, step by step. I’ll show you why testing the x-axis and y-axis is almost never enough (the “Two-Path Trap”), and why matching limits along ten different lines still proves absolutely nothing. Instead of guessing, we’ll build a reliable 4-step decision tree that tells you exactly what to do. You’ll learn when you can simply plug the numbers in, how to use the family of lines (y=mx) to disprove a limit instantly, when to switch to curved paths, and how to rigorously prove a limit exists using algebra or the Squeeze Theorem with polar coordinates. WANT MORE HELP? For full lessons, quizzes, and step-by-step practice: Calculus 3 (Multivariable Calculus) → https://courses.kristakingmath.com/library/calculus-3-236147/about/ WHAT YOU'LL LEARN - The 4-step decision tree for solving any multivariable limit - Why matching paths (like the axes) does not prove a limit exists - How to use the family of lines (y=mx) to disprove a limit quickly - The “Power-Match” tool for choosing curved paths - How to use algebraic simplification to show the limit exists (even at a discontinuity) - How to use the Polar Decision Rule to prove existence (or non-existence) CHAPTERS 0:46 — Step 1: Direct Substitution 1:40 — Step 2: Disproving with Paths: Testing the Axes 3:29 — Step 2: Disproving with Paths: Testing All Lines 5:58 — Step 2: Disproving with Paths: Testing Curves 7:53 — Step 3: How to Prove a Limit Exists (Algebra) 10:07 — Step 4: The Ultimate Proof (Polar Coordinates) 13:59 — Step 4: Polar Case 2 (The Limit Does Not Exist) 16:04 — (Challenge) Proving with Squeeze Theorem (Using a Famous Inequality) 23:06 — Your 4-Step Toolkit (Recap) CONNECT TikTok → https://www.tiktok.com/@kristakingmath Instagram → https://www.instagram.com/kristakingmath/ Facebook → https://www.facebook.com/kristakingmath/