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This is solution to assignment of Basic Mathematics course. This was submitted to Karunashankar Sidhu at Institute of Mathematical Sciences. It includes: Domain, Range, Function, Limit, Path, Value, Continuity, Ponit, Solution, AXis
Typology: Exercises
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MTH301 (Spring 2012)
Question # 1. Find the domain and range of
f x x
Solution : Domain of the function is x 0
Range of the function is 0 1 3
y
Question # 2. Evaluate the limit along the paths (a) the y -axis ( x = 0 ), (b) the x-axis ( y = 0 ), (c) the line y = mx for all value of m 0.
2 3 ( x y , lim) (0,0)^2
x y x y
Solution: (a) Along the line y-axis ( x = 0) 2 3 ( , ) (0,0)^2 3 ( , ) (0,0)^2
3 ( ) (0)^2
( ) (0)
lim
lim 0 0
lim
lim ( 0
x y
x y
y
y
x y x y y y
y y
y
(b) Along the line x-axis ( y = 0) 2 3 ( , ) (0,0)^2 2 ( , ) (0,0)^2
2 ( ) (0)^2
( ) (0)
lim
lim 0 0
lim
lim (
x y
x y
x
x
x y x y x x
x x
(c ) Along the line y = mx for all value of m 0 2 3 ( , ) (0,0)^2 2 3 ( , ) (0, )^2 2 3 ( , ) (0, )^2 3 ( , ) (0, )^2
2
lim
lim (^ ) ( )
lim (1^ ) (1 )
lim (1^ ) (1 )
x y
x y mx
x y mx
x y mx
x y x y x mx x mx x m x x m m x m
m
This shows that for different values of m there will different values of function.
Hence limit does not exist.
Question # 3. Determine whether the function is continuous at the stated point
2 2 2 2
x y (^) if x y f x y x y if x y
Solution: