Math 250 Test Paper on Limits and
Continuity of Multivariable Functions
University：California State University, Northridge
Course：Math 250 (Vector Calculus)
Instructor：David Klein
Test Duration：90 minutes
Full Score：100 points
I. Multiple Choice Questions (Each question worth 2 points, total 30
points)
What is the core criterion for determining the existence of a limit of a
multivariable function?（）
A. The limits as we approach the target point along the x-axis and y-axis
are equal
B. The limits as we approach the target point along any path within
the plane are equal
C. The limits as we approach the target point along two parallel lines are
equal
D. The function is defined at the target point
Substitution method is applicable for solving the limit of multivariable
functions under the premise that ()
A. The function is a polynomial function
B. The function is continuous at the target point
C. The limit of the function exists at the target point
D. The function value at the target point is not 0
When solving the limit of a multivariable function using polar
coordinate method, if the transformed result is related to θ, then ()
A. The limit exists, and the result is the transformed expression
B. The limit does not exist
, C. Further verification is required using the squeeze theorem
D. The extreme values are always 0
The following function has a limit at the point (0,0) ()
A. f(x,y) = (xy)/(x² + y²)
B. f(x,y) = (x³ + y³)/(x² + y²)
C. f(x,y) = (x² - y²)/(x² + y²)
D. f(x,y) = x³/(x⁴ + y²)
Three necessary conditions for the continuity of the function f(x,y) at the
point (x₀,y₀) do not include ().
A. The function is defined in some neighborhood of the point (x₀,y₀)
B. The function is differentiable at the point (x₀,y₀)
lim[(x,y)→(x₀,y₀)] C f(x,y) exists
D. lim[(x,y)→(x₀,y₀)] f(x,y) = f(x₀,y₀)
When using the squeeze theorem to solve the limit of a multivariable
function, the commonly used inequality is ()
A. xy ≥ (x² + y²)/2
B. sin t ≤ t
C. x ≥ √(x² + y²)
D. x + y ≥ x + y
Professor David recommended in class that when determining the non-
existence of limits of multivariable functions, the first paths to avoid
include ()
A. coordinate axes (y=0 or x=0)
B. lines y=kx (k is a constant)
C. The parabola y=kx² (k is a constant)
D. Curve y=sin x
Calculate lim[(x,y)→(1,2)] (2x²y - 3xy) to get ( )
A. 2
B. 4
C. 6
D. 8
Continuity of Multivariable Functions
University：California State University, Northridge
Course：Math 250 (Vector Calculus)
Instructor：David Klein
Test Duration：90 minutes
Full Score：100 points
I. Multiple Choice Questions (Each question worth 2 points, total 30
points)
What is the core criterion for determining the existence of a limit of a
multivariable function?（）
A. The limits as we approach the target point along the x-axis and y-axis
are equal
B. The limits as we approach the target point along any path within
the plane are equal
C. The limits as we approach the target point along two parallel lines are
equal
D. The function is defined at the target point
Substitution method is applicable for solving the limit of multivariable
functions under the premise that ()
A. The function is a polynomial function
B. The function is continuous at the target point
C. The limit of the function exists at the target point
D. The function value at the target point is not 0
When solving the limit of a multivariable function using polar
coordinate method, if the transformed result is related to θ, then ()
A. The limit exists, and the result is the transformed expression
B. The limit does not exist
, C. Further verification is required using the squeeze theorem
D. The extreme values are always 0
The following function has a limit at the point (0,0) ()
A. f(x,y) = (xy)/(x² + y²)
B. f(x,y) = (x³ + y³)/(x² + y²)
C. f(x,y) = (x² - y²)/(x² + y²)
D. f(x,y) = x³/(x⁴ + y²)
Three necessary conditions for the continuity of the function f(x,y) at the
point (x₀,y₀) do not include ().
A. The function is defined in some neighborhood of the point (x₀,y₀)
B. The function is differentiable at the point (x₀,y₀)
lim[(x,y)→(x₀,y₀)] C f(x,y) exists
D. lim[(x,y)→(x₀,y₀)] f(x,y) = f(x₀,y₀)
When using the squeeze theorem to solve the limit of a multivariable
function, the commonly used inequality is ()
A. xy ≥ (x² + y²)/2
B. sin t ≤ t
C. x ≥ √(x² + y²)
D. x + y ≥ x + y
Professor David recommended in class that when determining the non-
existence of limits of multivariable functions, the first paths to avoid
include ()
A. coordinate axes (y=0 or x=0)
B. lines y=kx (k is a constant)
C. The parabola y=kx² (k is a constant)
D. Curve y=sin x
Calculate lim[(x,y)→(1,2)] (2x²y - 3xy) to get ( )
A. 2
B. 4
C. 6
D. 8
