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📝 Limits of multivariable functions along curves (13 MCQs)

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

←Previous Topic:Limits and Continuity in Partial Derivatives📝 Limits of multivariable functions along curvesOpen and closed sets in multivariable calculus:Next Topic→

What is Limits of multivariable functions along curves?

Definition:
Evaluating $\lim_{t\to 0} f(x(t), y(t))$ along specific parametric paths $(x(t), y(t)) \to (a,b)$ to test for limit existence.

Example:
Approaching $(0,0)$ along $y = mx$ yields different values for $f(x,y) = \frac{xy}{x^2+y^2}$ depending on slope $m$ .

Reason:
If limits differ along distinct paths, the general limit does not exist; this negative test is often easier than proving existence directly.

Easy

Medium

Hard

📝 All Limits of multivariable functions along curves MCQs

Q1. A function $f(x,y)$ approaches 3 along every straight line through the origin, but equals $\frac{2x^4}{x^4+y^2}$ elsewhere. A student concludes the limit at (0,0) is 3. Which statement best critiques this reasoning?

A.The conclusion is valid because lines form a basis for all paths.

B.The conclusion is invalid; testing only lines is insufficient as parabolic paths may yield different limits. ✅

C.The conclusion is valid since the function is continuous along radial directions.

D.The conclusion is invalid because the function is undefined at the origin.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Testing limits solely along straight lines is a common misconception. Even if all linear paths agree, nonlinear curves like parabolas can produce different limiting values. Higher-order analysis requires examining families of curves beyond lines to confirm or refute existence of a multivariable limit rigorously.

Q2. Consider $f(x,y)=\frac{x^2y}{x^4+y^2}$ . Along which curve does the limit as $(x,y)\to(0,0)$ differ from the limit along $y=x$ ?

y=x^2

✅

y=0

y=-x

y=x^3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Along $y=x$ , the expression simplifies to zero. However, substituting $y=x^2$ yields $\frac{x^4}{x^4+x^4}=1/2$ , demonstrating path dependence. This application problem tests understanding that agreement on some paths does not guarantee a unique limit, requiring strategic curve selection.

Q3. Given a contour plot where level curves near the origin appear to converge radially but spacing changes asymmetrically in quadrants II and IV, what can be inferred about $\lim_{(x,y)\to(0,0)}f(x,y)$ ?

A.The limit exists and equals the central contour value.

B.The limit likely does not exist due to directional asymmetry suggesting path-dependent behavior. ✅

C.The limit exists because contours are closed curves.

D.Cannot determine without an explicit formula.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Graph interpretation reveals that asymmetric contour spacing indicates varying rates of approach from different directions. Even with radial appearance, non-uniformity across quadrants suggests the function values do not stabilize uniformly, implying the limit may fail to exist despite visual convergence cues.

Q4. A student evaluates $\lim_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^4}$ by converting to polar coordinates and obtaining $r\cos\theta\sin^2\theta$ , concluding the limit is 0. Identify the flaw.

A.Polar conversion always validates limits when the result is independent of

\theta

B.The algebraic simplification incorrectly canceled terms; the denominator becomes

r^2(\cos^2\theta+r^2\sin^4\theta)

, not

r^2

. ✅

C.The limit actually equals 1 along certain paths.

D.Polar coordinates cannot handle rational functions with mixed degrees.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Error analysis shows misapplication of polar substitution. The denominator $x^2+y^4$ does not factor cleanly as $r^2$ times a theta-only function due to unequal powers. This creates hidden path dependence that polar form obscures, making direct curve testing necessary for correct evaluation.

Q5. For modeling heat diffusion near a point source, temperature $T(x,y)$ must have a well-defined limit at the origin. If $T=\frac{x^3-y^3}{x^2+y^2}$ along measurement paths, which additional test ensures physical consistency?

A.Verify continuity along coordinate axes only.

B.Check boundedness using squeeze theorem with

|T|\leq\sqrt{x^2+y^2}

. ✅

C.Confirm limit agreement along logarithmic spirals.

D.Test only along paths where

x=y

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Conceptual understanding requires recognizing that physical models demand unique limits. While specific curves help disprove existence, proving existence needs uniform bounds. The squeeze theorem provides rigorous confirmation independent of path choice, ensuring the mathematical model reflects physically meaningful continuous behavior at critical points.

Q6. If $f(x,y)\to L$ along every polynomial curve $y=p(x)$ through the origin, does $\lim_{(x,y)\to(0,0)}f(x,y)=L$ necessarily hold?

A.Yes, polynomials are dense in continuous functions.

B.No, there exist non-algebraic paths yielding different limits even when all polynomial paths agree. ✅

C.Yes, because Taylor expansions approximate any smooth path.

D.No, unless the function is also differentiable at the origin.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This Olympiad-style question probes deep topological understanding. Polynomial curves, while rich, do not exhaust all possible approaches. Pathological functions can be constructed to agree on all algebraic curves yet diverge along transcendental paths, demonstrating that even infinite families of curves cannot substitute for epsilon-delta proof of limit existence.

Q7. Two students analyze $g(x,y)=\frac{x^2-y^2}{x^2+y^2}$ . Student A claims no limit exists because values range [-1,1]. Student B argues the limit is 0 along $y=x$ . Who demonstrates better higher-order reasoning?

A.Student A, because identifying the full range proves non-existence comprehensively.

B.Student B, because finding one valid path establishes the limit.

C.Neither; both miss that the limit exists and equals 0.

D.Student A partially, but should specify distinct limiting values along different curves rather than just the range. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Mixed concepts require distinguishing between range and path-specific limits. Student A correctly identifies non-existence but lacks precision; merely stating the range doesn't prove path dependence. Better reasoning explicitly shows $y=0$ gives 1 while $x=0$ gives -1, directly demonstrating conflicting limits along specific curves.

Q8. When evaluating $\lim_{(x,y)\to(0,0)}\frac{x^ay^b}{x^c+y^d}$ , under what condition does choosing $y=x^{c/d}$ guarantee revealing non-existence if the limit fails?

A.When

a+b>c

and

d>0

B.When the numerator's weighted degree exceeds the denominator's homogeneity after substitution. ✅

C.Always, regardless of exponents.

D.Only when

a=b

and

c=d

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Application requires understanding scaling balance. Substituting $y=x^{c/d}$ equalizes denominator terms, making the expression homogeneous in x. If the resulting power of x in the numerator differs from the denominator's effective degree, the limit depends on the exponent relationship, systematically exposing path dependence through strategic curve selection.

Q9. A function satisfies $|f(x,y)|\leq\frac{|x|^3+|y|^3}{x^2+y^2}$ . A peer claims the limit is 0 based solely on testing $y=mx$ . What crucial step was omitted?

A.Verifying the inequality holds for all

(x,y) \neq (0,0)

B.Applying the squeeze theorem to bound the expression independently of path. ✅

C.Testing additional curves like

y=x^2

D.Converting to polar coordinates first.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Direct recall of methodology shows testing paths alone cannot prove limits. The given inequality enables squeeze theorem application: since $\frac{|x|^3+|y|^3}{x^2+y^2}\leq|x|+|y|\to0$ , the bound confirms the limit universally. Omitting this rigorous justification renders path-based evidence insufficient despite correct intuition.

Q10. In optimizing a surface $z=f(x,y)$ near a critical point, you suspect the limit doesn't exist. After finding two curves with different limits, what further analysis strengthens your conclusion for engineering applications?

A.Compute partial derivatives to check differentiability.

B.Quantify the rate of divergence along each path to assess sensitivity. ✅

C.Find a third curve matching one of the previous limits.

D.Prove continuity along all circular arcs.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Scenario-based reasoning extends beyond pure mathematics. Engineering contexts require understanding not just existence but stability. Quantifying divergence rates informs tolerance specifications and numerical method reliability. This transforms abstract non-existence into actionable insight about system behavior near singularities, bridging theoretical analysis with practical design constraints.

Q11. Which statement correctly distinguishes between 'limit along a curve' and 'multivariable limit'?

A.They are equivalent if the curve passes through the point.

B.The former is a restricted evaluation; the latter requires uniformity across all possible approaches. ✅

C.Both depend solely on function values at the point.

D.Multivariable limits ignore curved paths entirely.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Conceptual understanding clarifies foundational definitions. Limits along curves examine behavior constrained to specific trajectories, serving as necessary but insufficient conditions. True multivariable limits demand identical convergence regardless of approach path, embodying a stronger uniformity condition. Confusing these leads to erroneous conclusions about continuity and differentiability in higher dimensions.

Q12. Given $h(x,y)=\frac{\sin(xy)}{x^2+y^2}$ , a graph shows oscillations damping toward origin along axes but persistent ripples along $y=1/x$ away from origin. How should this inform limit analysis at (0,0)?

A.Ripples far from origin are irrelevant; focus on local behavior near (0,0). ✅

B.The function has no limit due to global oscillatory behavior.

C.Local damping suggests limit exists; distant ripples don't affect pointwise limit.

D.Graph is misleading; analytical methods override visual evidence.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Graph-based interpretation requires distinguishing local versus global features. Limit analysis concerns arbitrarily small neighborhoods around the point. While $y=1/x$ exhibits oscillations, this path doesn't approach (0,0). Recognizing relevant domain restrictions prevents misinterpreting asymptotic behavior as evidence against local limit existence, emphasizing precise spatial reasoning.

Q13. Suppose $f(x,y)\to5$ along all curves $y=kx^n$ for integer $n\geq1$ , but $f(x,x^2\sin(1/x))\to7$ . What does this reveal about sufficient conditions for limit existence?

A.Agreement on monomial curves guarantees existence for smooth functions.

B.No countable family of curves suffices; uncountably many pathological paths may violate convergence. ✅

C.The limit exists because most standard curves agree.

D.Differentiability would resolve the discrepancy.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: This challenging problem exposes limitations of curve-testing heuristics. Even infinite parametric families cannot capture all possible approaches. The oscillatory curve exploits rapid sign changes invisible to power-law paths, demonstrating that limit existence fundamentally requires topological arguments beyond sequential or parametric verification, highlighting the gap between intuitive testing and rigorous proof.

🔗 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:Limits and Continuity in Partial Derivatives📝 Limits of multivariable functions along curvesOpen and closed sets in multivariable calculus:Next Topic→