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Calculus IIIChapter 2: Partial DerivativesSection 2-1

Limits — Question 9

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Question 9

Suppose (a,b)(a,b) lies strictly inside the disk x2+y2<25x^2+y^2<25. Prove lim(x,y)→(a,b)25−x2−y2=25−a2−b2,\lim_{(x,y)\to(a,b)}\sqrt{25-x^2-y^2}=\sqrt{25-a^2-b^2}, and give a concrete neighborhood that remains inside the domain.

Tasks

  1. Find a positive safety margin to the boundary.

  2. Choose a disk around (a,b)(a,b) contained in the domain.

  3. Justify the limit by composition.

Original worksheet page 1: question and worked solution for 2-1-009
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Question 9 – Solution

Strategy. Use the triangle inequality to keep nearby points inside the radius-55 disk, then apply continuity.

See the diagram in the original worksheet below.

Step 1: Margin Let s=a2+b2<5s=\sqrt{a^2+b^2}<5. The radial margin is 5−s>05-s>0.

Step 2: Neighborhood Choose δ0=5−s2.\boxed{\delta_0=\frac{5-s}{2}}. If (x−a)2+(y−b)2<δ0\sqrt{(x-a)^2+(y-b)^2}<\delta_0, then by the triangle inequality x2+y2≤s+(x−a)2+(y−b)2<s+5−s2<5.\sqrt{x^2+y^2}\le s+\sqrt{(x-a)^2+(y-b)^2}<s+\frac{5-s}{2}<5. Thus this entire neighborhood lies in the square-root domain.

Step 3: Composition The polynomial 25−x2−y225-x^2-y^2 is continuous and remains positive there; ⋅\sqrt{\cdot} is continuous on positive inputs. Their composition is continuous at (a,b)(a,b), proving the boxed limit stated in the question.

Original worksheet page 2: question and worked solution for 2-1-009

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