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
The correct expression for ρ in the polar coordinate transformation of
the limit of a multivariable function is ()
A. ρ = (x - x₀)² + (y - y₀)²
B. ρ = √[(x - x₀) + (y - y₀)]
C. ρ = √[(x - x₀)² + (y - y₀)²]
D. ρ = x - x₀ + y - y₀
function f(x,y) = (x² - y²)/(x - y) is discontinuous at the point (1,1) due to
the direct reason ()
A. The limit of the function at this point does not exist
B. The function is undefined at this point
C. The limit of the function at this point is not equal to the function
value
D. The function value at this point is 0
if lim[(x,y)→(0,0)] f(x,y) = 0, and f(0,0) = 1, then the function f(x,y) at the
point (0,0) is ()
A. continuous
B. removable discontinuity
C. infinite discontinuity
D. The limit does not exist
Calculate lim[(x,y)→(0,π/3)] (x sin y + y cos x) to get ()
A. π/2
B. π/3
C. π/4
D. π/6
About the limits and continuity of multivariable functions, which of the
following statements is correct ()
A. Existence of limit → Function is continuous
B. Function is continuous → Existence of limit (and equal to the function
value)
C. Non-existence of limit → Function is continuous
D. Function is continuous → Non-existence of limit
Using polar coordinates, the limit [((x,y)→(0,0))] (x⁴ + 2x²y² + y⁴)/(x² +
y²) is calculated to be ()
A. 0
B. 1
C. 2
D. Does not exist
If the limit of a multivariable function does not exist at a point, then the
function at that point ()
A. is definitely continuous
B. is definitely discontinuous
C. May be continuous
D. Cannot determine continuity
II. True/False Questions (1 point each, total 10 points)
In a multivariable function, if the limit is the same along any path in the
plane as it approaches a point, then the limit at that point exists. (）
Only by verifying that the limits approaching a certain point along the
two lines y=2x and y=3x are equal can we determine that the limit of
the multivariable function exists at that point.）
The composite functions of elementary functions (polynomials,
trigonometric functions, exponential and logarithmic functions) are
continuous within their domains.）
The polar coordinate method is applicable to all scenarios of solving
limits of multivariable functions.）
If a multivariable function is continuous at a certain point, then the limit
at that point must exist, and the limit value is equal to the function
value.）
If the limit of a multivariable function approaches infinity along a certain
path as it approaches a point, then it can be directly determined that
the limit of the function at that point does not exist.）
Substitution can be directly used for the calculation of limits of all
multivariable functions.）
The function f(x,y) = { (x²y)/(x² + y²) , (x,y)≠(0,0)；0 , (x,y)=(0,0) } is
continuous at the point (0,0).）
lim[(x,y)→(0,0)] (x²y²)/(x² + y²) = 0。（ ）
The three conditions for the continuity of a multivariable function
(defined, limit exists, limit equals function value) are all essential;
missing any one of them makes the function discontinuous at that
point.）
III. Calculation Problems (5 points each, total 30 points)
Calculate lim[(x,y)→(2,3)] (3x² - 2xy + y²) using the substitution method.
Solve lim[(x,y)→(0,0)] (x³ + y³)/(x² + y²) using polar coordinates.
Determine whether the limit lim [(x,y)→(0,0)] (xy)/(x² + y²) exists. If it
does not exist, explain the reason.
Use the squeeze theorem to solve lim [(x,y)→(0,0)] [sin(xy)]/√(x² + y²).
Discuss the continuity of the function f(x,y) = { sin(xy)/(xy) , xy≠0；1 ,
xy=0 } at the point (0,0).
Given that the function f(x,y) = { (x² + y²)/[√(x² + y² + 1) - 1] ,
(x,y)≠(0,0)；k , (x,y)=(0,0) } is continuous at the point (0,0), find the value
of k.
IV. Problem Solving (10 points per question, total 30 points)
Compare the essential differences between the limits of single-variable
functions and multivariable functions, and illustrate how this difference
affects limit determination with specific examples.
Given the function f(x,y) = { (x²y)/(x² + y²) , (x,y)≠(0,0)；a , (x,y)=(0,0) }，
（1）求lim[(x,y)→(0,0)] f(x,y)；
(2) Determine the value of a to make the function f(x,y) continuous at
the point (0,0);
(3) If a=2, determine the type of discontinuity of the function f(x,y) at
the point (0,0) and explain the reason.
Use two different methods (such as the path method, polar coordinate
method) to determine whether lim [(x,y)→(0,0)] (x²y)/(x⁴ + y²) exists,
write detailed steps and conclusions.
