← Back to 14. Partial Derivatives Calculus

📝 Multivariable limits at discontinuities (13 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 13 questions available

←Previous Topic:Continuity of multivariable functions📝 Multivariable limits at discontinuitiesContinuity at boundary points in multivariable:Next Topic→

What is Multivariable limits at discontinuities?

Definition:
Analysis of behavior near points where $f$ is undefined or limit differs from function value, often involving asymptotes or removable singularities.

Example:
At $(0,0)$ , $f(x,y) = \frac{\sin(x^2+y^2)}{x^2+y^2}$ has a removable discontinuity with limit 1 despite being undefined at origin.

Reason:
Identifying discontinuity types guides redefinition strategies and reveals where derivative-based approximations will fail or require special handling.

Easy

Medium

Hard

📝 All Multivariable limits at discontinuities MCQs

Q1. A multivariable function $f(x,y)$ is defined as $\frac{x^2 y}{x^4 + y^2}$ for $(x,y) \neq (0,0)$ and 0 at the origin. A student claims the limit exists because approaching along $y=mx$ yields 0 for all $m$ . Which analysis correctly identifies the flaw in this reasoning?

A.The student correctly applied the definition of continuity since linear paths are sufficient for polynomial denominators.

B.The limit does not exist because approaching along the parabolic path

y=x^2

yields

\frac{1}{2}

, contradicting the linear path result. ✅

C.The function is continuous at the origin because the squeeze theorem applies with bounding functions

|x|

and

|y|

D.The error lies in arithmetic; substituting

y=mx

actually yields

\frac{m}{1+m^2}

, which depends on

m

, proving non-existence.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This question targets error analysis and higher-order thinking by requiring students to recognize that verifying linear paths is insufficient for multivariable limits. The specific parabolic path $y=x^2$ exposes the discontinuity hidden from linear approaches, demonstrating why comprehensive path testing or polar coordinate analysis is necessary for rigorous verification.

Q2. Consider a surface modeling heat distribution where temperature $T(x,y)$ has a removable discontinuity at a sensor location $(a,b)$ . If $\lim_{(x,y)\to(a,b)} T(x,y) = L$ but $T(a,b)$ is undefined due to sensor failure, which statement best describes the physical implication for interpolation?

A.Interpolation is impossible because the function lacks a defined value at the critical point, making the model invalid.

B.The missing sensor data can be reliably replaced by

L

because the surrounding thermal field converges smoothly to this unique value. ✅

C.Only directional derivatives can be estimated, but the scalar temperature value remains fundamentally unknowable without new hardware.

D.The limit

L

represents an average of neighboring values but cannot guarantee local continuity required for differential equations.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This scenario-based application question connects abstract limit concepts to real-world sensor modeling. Students must distinguish between mathematical existence of a limit and physical measurability, understanding that removable discontinuities allow reliable data reconstruction through limiting processes, which is essential for engineering applications involving incomplete datasets.

Q3. Given the contour plot of $f(x,y)$ showing increasingly dense level curves near the origin with alternating positive and negative values, but no labeled numerical values, what can be definitively concluded about $\lim_{(x,y)\to(0,0)} f(x,y)$ ?

A.The limit exists and equals zero because contour density indicates convergence to a single point.

B.The limit does not exist because alternating signs and increasing density suggest oscillatory behavior preventing convergence. ✅

C.The limit exists but cannot be determined numerically without explicit function definition or scale markers.

D.No conclusion can be drawn because contour plots only represent differentiable regions and fail at discontinuities.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This graph-based interpretation question requires analyzing visual topographical data rather than computing algebraically. Dense alternating contours indicate rapid sign changes characteristic of oscillatory discontinuities like $\sin(1/r)$ , helping students develop intuition connecting geometric representations to analytical limit behavior without relying solely on symbolic manipulation.

Q4. When evaluating $\lim_{(x,y)\to(0,0)} \frac{x^3 - y^3}{x^2 + y^2}$ , which method provides the most efficient rigorous proof of existence while avoiding common pitfalls associated with path-dependent testing?

A.Testing paths

y=mx

x=0

, and

y=x^2

to show consistent results across diverse approaches.

B.Converting to polar coordinates yielding

r(\cos^3\theta - \sin^3\theta)

and applying the squeeze theorem as

r\to 0

. ✅

C.Applying L'Hôpital's rule separately to numerator and denominator partial derivatives until determinate form emerges.

D.Factoring the numerator as difference of cubes and canceling common terms with the denominator directly.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This comparative methods question emphasizes strategic selection of proof techniques. Polar conversion transforms two-variable limits into single-variable problems with bounded angular components, providing systematic rigor superior to inconclusive path testing. Students learn when coordinate transformation outperforms direct substitution or factoring, developing metacognitive awareness of method appropriateness.

Q5. A student computes $\lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y^2}$ using polar coordinates, obtaining $\frac{\sin(2\theta)}{2}$ , then concludes the limit is 0 because sine is bounded. Identify the precise logical error in this argument.

A.Polar coordinates cannot be used for rational functions with homogeneous denominators of equal degree.

B.Boundedness of

\sin(2\theta)

is irrelevant; the expression still depends on

\theta

, violating the requirement for uniform convergence independent of direction. ✅

C.The student should have converted to spherical coordinates since two-dimensional polar fails for this specific rational form.

D.Arithmetic error occurred; the correct polar form is

\sin(\theta)\cos(\theta)

which approaches infinity, not zero.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This error analysis question targets a sophisticated misconception about polar coordinate limits. Students often confuse boundedness with convergence, failing to recognize that dependence on $\theta$ means different approach angles yield different limiting values. Understanding uniform independence from direction is crucial for distinguishing true limits from merely bounded expressions.

Q6. For the piecewise function $f(x,y) = 1$ if $y > x^2$ and $f(x,y) = 0$ otherwise, analyze the behavior at $(0,0)$ considering both topological and sequential definitions of limits.

A.The limit exists and equals 0 because the parabola has measure zero in any neighborhood of the origin.

B.The limit does not exist because every neighborhood contains points satisfying both conditions, preventing unique sequential convergence. ✅

C.The limit exists and equals 1 because the region

y>x^2

occupies larger area near the origin.

D.Sequential limits exist individually along vertical and horizontal axes but the overall limit remains undefined due to asymmetry.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This challenging mixed-concepts question integrates topology, measure theory intuition, and sequential characterization. Students must understand that limit existence requires consistency across ALL sequences, not just measure-theoretic dominance or axis-aligned approaches. The parabolic boundary creates unavoidable ambiguity in every neighborhood, illustrating subtle distinctions between almost-everywhere properties and strict limit definitions.

Q7. In optimization problems, encountering $\lim_{(x,y)\to(0,0)} \frac{x^2 y^2}{x^4 + y^4}$ requires determining continuity before applying gradient methods. What is the most appropriate next step after confirming the limit exists via AM-GM inequality?

A.Immediately compute partial derivatives using quotient rule since continuity guarantees differentiability.

B.Verify whether the extended function definition at origin matches the computed limit value before proceeding. ✅

C.Apply second derivative test directly since limit existence implies smoothness sufficient for classification.

D.Transform variables to

u=x^2, v=y^2

to simplify subsequent differentiation steps.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This application question bridges limit theory with practical optimization workflow. Students often conflate continuity with differentiability or assume automatic smoothness. Recognizing that limit computation alone doesn't ensure function definition matches the limit prevents erroneous gradient calculations, emphasizing the distinction between limit existence and proper function extension in applied contexts.

Q8. Compare the discontinuity types of $f(x,y)=\frac{x^2-y^2}{x^2+y^2}$ and $g(x,y)=\frac{x^3}{x^2+y^2}$ at the origin. Which characterization accurately distinguishes their behaviors?

A.Both exhibit essential discontinuities with unbounded oscillation preventing any meaningful extension.

B.f has a jump-type directional discontinuity while g has a removable discontinuity allowing continuous extension. ✅

C.Both are removable since polar forms show bounded numerators vanishing as r approaches zero.

D.f is removable but g exhibits essential discontinuity due to cubic numerator growth rate.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This comparative conceptual question requires classifying discontinuity severity beyond binary existence. Students must recognize that directional dependence in f creates irreconcilable approach differences (jump-like), while g's higher-order numerator enables squeezing despite similar denominator structure. Distinguishing removable versus essential discontinuities informs appropriate remediation strategies in analysis and modeling.

Q9. A physics model predicts particle velocity $v(x,y) = \frac{x^3 y}{x^6 + y^2}$ near collision point. Numerical simulations along grid lines suggest convergence to 0, but theoretical analysis reveals issues. What explains this discrepancy between computational and analytical results?

A.Numerical precision errors accumulate near origin causing false convergence appearance in discrete sampling.

B.Grid-aligned sampling misses the critical path

y=x^3

where the function equals

\frac{1}{2}

, exposing hidden discontinuity. ✅

C.The model itself is physically invalid since velocities cannot depend on sixth-power spatial terms.

D.Simulations correctly capture the true limit; analytical methods overestimate complexity for this rational function.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This scenario-based error analysis highlights dangers of numerical verification without theoretical grounding. Discrete grids systematically miss pathological curves like $y=x^3$ that reveal discontinuities invisible to axis-aligned sampling. Students learn computational methods complement but never replace analytical rigor, especially for functions with high-degree homogeneity creating deceptive numerical artifacts.

Q10. When proving $\lim_{(x,y)\to(0,0)} \frac{x^4 y}{x^4 + y^4} = 0$ , a student writes: 'Since $|x^4 y| \leq x^4 |y|$ and denominator $\geq x^4$ , the fraction $\leq |y| \to 0$ .' Evaluate this proof's validity.

A.Valid because the inequality chain correctly bounds the expression by a quantity approaching zero. ✅

B.Invalid because dividing by

x^4

assumes

x \neq 0

, requiring separate treatment of the y-axis approach.

C.Invalid because

x^4 + y^4 \geq x^4

is false when

y=0

, breaking the inequality foundation.

D.Valid but inefficient; polar coordinates would provide simpler verification without case analysis.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This proof evaluation question develops critical assessment skills for epsilon-delta arguments. While the student's reasoning appears sound, careful examination reveals the inequality holds universally since $y^4 \geq 0$ always. Students practice validating logical structure rather than just conclusions, recognizing when apparent flaws are actually acceptable simplifications in rigorous proofs.

Q11. Consider $h(x,y) = \begin{cases} \frac{x^n y^m}{x^p + y^q} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \end{cases}$ . For which relationship between exponents does continuity at origin hold regardless of specific positive integer values?

A.Continuity requires

n+m > \max(p,q)

ensuring numerator dominates denominator growth.

B.Continuity holds when

\frac{n}{p} + \frac{m}{q} > 1

based on weighted homogeneity scaling. ✅

C.Continuity demands

n \geq p

AND

m \geq q

simultaneously for component-wise domination.

D.Continuity occurs only if

n+m > p+q

matching total degree comparison.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This Olympiad-style generalization question abstracts specific examples into parametric conditions. Students derive scaling relationships through dimensional analysis or weighted polar substitution, recognizing that simple degree comparisons fail for mixed exponents. The weighted condition captures nuanced balance between anisotropic growth rates, representing deep structural understanding beyond memorized special cases.

Q12. Two students debate whether $\lim_{(x,y)\to(0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2}$ requires multivariable techniques. Student A insists on polar conversion; Student B argues single-variable substitution $u=x^2+y^2$ suffices. Who is correct and why?

A.Student A because multivariable limits always require coordinate transformation for rigor.

B.Student B because the expression depends solely on radial distance, reducing to single-variable limit

\lim_{u\to 0^+} \frac{\sin u}{u}

. ✅

C.Neither; the limit does not exist due to oscillatory sine behavior near origin.

D.Both are equally valid but Student A's method is preferred for pedagogical completeness.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: This conceptual comparison question identifies when multivariable problems reduce to single-variable analysis. Recognizing radial symmetry allows legitimate dimension reduction, avoiding unnecessary complexity. Students learn to detect functional dependencies that simplify analysis, distinguishing genuine multivariable challenges from disguised univariate problems, optimizing problem-solving efficiency through structural insight.

Q13. An engineer models stress concentration as $\sigma(r,\theta) = \frac{\cos(3\theta)}{r^\alpha}$ near crack tip. For material safety, the stress must remain bounded as $r\to 0$ . Analyze the implications of choosing $\alpha = 0$ versus $\alpha > 0$ regarding limit existence and physical plausibility.

A.α=0 gives bounded oscillatory limit depending on direction; α>0 causes unbounded blow-up violating material strength limits.

B.α=0 produces no limit due to angular dependence; α>0 yields infinite stress incompatible with continuum mechanics assumptions. ✅

C.Both choices produce existing limits but only α=0 satisfies energy integrability requirements for fracture models.

D.α>0 ensures limit exists and equals zero; α=0 creates removable discontinuity fixable by redefinition.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: This advanced application integrates mathematical limits with engineering constraints. Students analyze how exponent choice affects both mathematical existence and physical admissibility, recognizing that boundedness alone doesn't guarantee meaningful limits when directional dependence persists. Connecting abstract convergence criteria to material failure thresholds demonstrates synthesis of mathematical rigor with domain-specific plausibility requirements.

🔗 Related Topics (MCQs)

📝 Functions of Two or More Variables MCQs 📝 Multivariable function notation and terminology MCQs 📝 Functions of two variables from tables MCQs 📝 Level curves and contour plots MCQs 📝 Graphing functions of two variables MCQs

←Previous Topic:Continuity of multivariable functions📝 Multivariable limits at discontinuitiesContinuity at boundary points in multivariable:Next Topic→