📝 General vs path limits multivariable (13 MCQs)

📖 Explanation: Agreement along all straight lines is necessary but insufficient for the existence of a general multivariable limit. A counterexample such as $f(x,y)=\frac{x^2y}{x^4+y^2}$ yields zero along every line yet approaches different values along $y=x^2$, demonstrating that smooth nonlinear curves can reveal path dependence invisible to linear analysis.

📖 Explanation: While $f(t,g(t)) \to 0$ for all smooth $g$ with $g(0)=0$, this alone doesn't constitute proof of the general limit. However, using polar coordinates confirms the limit is indeed 0. The key insight is that even comprehensive smooth-curve testing requires supplementary global bounding arguments to establish existence rigorously in multivariable calculus.

📖 Explanation: Denser level curves along $y=x^3$ indicate rapid change or oscillation specifically on that smooth path, suggesting $f$ may approach different values or fail to stabilize there. Even if other directions appear well-behaved, a single smooth curve exhibiting pathological behavior can prevent the general limit from existing, highlighting the importance of examining non-obvious trajectories in graphical analysis.

📖 Explanation: Monomial paths $y=x^k$ form a countable set within the uncountable space of smooth curves. Functions can be engineered to agree on all algebraic paths yet diverge along flat functions like $y=e^{-1/x^2}$, which vanish faster than any polynomial. This illustrates that verifying limits requires either universal bounds or testing beyond standard algebraic families, emphasizing the subtlety of multivariable continuity.

📖 Explanation: Although $T$ appears radially symmetric, the cooling rate $\partial T/\partial t$ involves second-order spatial derivatives sensitive to path curvature. Approaching along curved trajectories captures how gradient alignment with thermal flux varies, which pure radial analysis misses. In physical modeling, assuming isotropy based solely on radial limits can lead to incorrect predictions about transient behavior near singularities.

📖 Explanation: The path $x=y^4$ gives $\frac{y^4 \cdot y^2}{y^8 + y^4} = \frac{y^6}{y^4(y^4+1)} \to 0$, actually still zero. But consider $x=y^2$: yields $\frac{y^2 \cdot y^2}{y^4 + y^4} = \frac{1}{2}$. Wait—rechecking: $x=y^2$ gives numerator $y^2 \cdot y^2 = y^4$, denominator $y^4 + y^4 = 2y^4$, so limit $1/2 \neq 0$. Thus testing $x=y^2$ should have revealed nonexistence. The real error is incomplete path selection; $x=y^2$ is smooth and exposes the flaw, showing that even common nonlinear tests can miss critical behaviors if not chosen strategically.

📖 Explanation: Limit existence depends solely on behavior near, not at, the point. Undefinedness at the origin doesn't affect the limit. Since all smooth curves with nonzero tangents cover all possible approach directions (including vertical via reparameterization), agreement implies the general limit exists. This reinforces that limits concern neighborhoods, not point values, and smoothness with nonzero derivative ensures comprehensive directional coverage.

📖 Explanation: Researcher B identified a specific smooth parabolic path yielding $1/2 \neq 0$, proving the general limit does not exist. Lines alone gave a false sense of convergence. This exemplifies why HOTS requires moving beyond basic directional tests: smooth nonlinear curves can expose hidden dependencies that linear approximations mask, making them essential diagnostic tools in multivariable limit analysis.

📖 Explanation: Homogeneity of degree 0 means $f(r\cos\theta, r\sin\theta) = f(\cos\theta, \sin\theta)$, so the function depends only on angle. If the angular limit $L$ is constant for all $\alpha$, then $f$ is constant on the punctured plane, ensuring the general limit exists. However, if $L$ varied with $\alpha$, the limit wouldn't exist. The key is recognizing that homogeneity collapses dimensionality, making angular uniformity equivalent to full limit existence.

📖 Explanation: Numerical sampling, even extensive, cannot guarantee limit existence because the space of smooth curves is infinite-dimensional. An adversarially constructed function could agree with 3 on sampled paths but deviate on unsampled ones. While suggestive, such evidence lacks mathematical certainty. This highlights the distinction between empirical plausibility and rigorous proof in analysis, emphasizing that HOTS requires understanding the limitations of computational verification versus theoretical justification.

📖 Explanation: Although $f \to 0$ along all power-law curves $y=cx^n$, one can construct a smooth curve like $y = x^3 \sin(1/x)$ that oscillates while approaching zero, potentially yielding nonzero limits due to resonance with the denominator's structure. Such transcendental curves lie outside algebraic families, demonstrating that completeness of path testing requires more than dense subsets—it demands either global estimation or explicit construction of counterexamples within the smooth category.

📖 Explanation: Pointwise agreement along each smooth curve doesn't suffice; uniformity across the family of curves is required to interchange limits. Without uniform control, different curves could converge arbitrarily slowly, allowing escape from any proposed limit bound. This connects to advanced concepts in topology and functional analysis, illustrating that HOTS at the Olympiad level demands synthesizing limit definitions with compactness or equicontinuity principles beyond standard calculus curriculum.

📖 Explanation: The original statement provides a necessary condition: existence implies path agreement. Reversing it falsely treats path agreement as sufficient. This is a classic logical error in multivariable calculus where students conflate 'if P then Q' with 'if Q then P'. Recognizing this distinction is foundational HOTS, as many limit nonexistence proofs rely precisely on finding a single violating curve, while existence requires stronger global arguments beyond pathwise verification.