2025-26 Second Semester
MATH 2822 Mathematical Methods for Actuarial Science II
Assignment 3
2025/02/10
Name:
University No:
1. Evaluate the following limits, or show that the limit does not exist.
(a)
lim
(x,y)→(1,1)
(b)
lim
(x,y)→(0,0)
(c)
√
x−y
√
x− y
x2 sin
1
x2 − y 2
xy 2
(x,y)→(0,0) x2 + y 4
lim
2. Let f : R2 \ {(1, 0)} ∪ {(0, y) : y < 0} → R be a function defined by


|x|y
f (x, y) = (x − 1)2 + |ey − 1|

1
if (x, y) ∈
/ {(0, 0), (0, 1)}
otherwise
Find all points in the domain of f at which f is continuous.
.


xy
2
2
3. Define f : R2 → R by f (x, y) = x + y
0
(a) Fvaluate fx and fy .
(b) Is f differentiable? Explain.
if (x, y) ̸= (0, 0)
if (x, y) = (0, 0).
4. Suppose Newton’s method is used to solve the system of equations
2
x + y2 = 5
x3 + y 3 = 2
x
2
with initial guess 0 =
. Find y2 .
y0
−1
(Round your answer to 4 decimal places.)
5. (a) Find the second Taylor polynomial of f (x, y) = e2x cos y at (0, π2 ), and use it to find a
quadratic approximation of the value of f (0.1, 1.5). Round your answer to four decimal
places.
π
(b) Find an upper bound for the error of the approximation if |x| ≤ 0.2 and y −
≤ 0.1.
2
(Round your answer to 4 decimal places.)