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**Unformatted text preview: **Math 1920, Prelim I Practice Problems
1. Suppose that f (x, y ) is a function deﬁned and continuous everywhere in the plane
except at (0, 0). The ﬁgure shows some of the distinct level curves as solid circles
tangent to the x-axis at the origin. Suppose further that lim f (0, y ) = 0 = lim f (x, 0).
y →0 Does lim (x,y )→(0,0) x→0 f (x, y ) exist? Does lim f (x, kx) = 0 for all k ?
x→0 y
f(x,y) = 1 x 2. Is there a non-zero value of the number a such that f (x, y ) = (x2 + y 2 )a satisﬁes
Laplace’s equation
∂2f
∂2f
+ 2 = 0.
∂x2
∂y
3. (a) Show that the vectors i, j, k, i + j + k form the vertices of a regular tetrahedron.
(b) Which is larger, the edges of a regular tetrahedron, or the distance between opposite edges? What is the ratio of those distances? ...

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