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Limits Part 1-Basic Mathematics-Assignment Solution, Exercises of Mathematics

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

2011/2012

Uploaded on 08/03/2012

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Solution of Assignment # 1
MTH301 (Spring 2012)
Question # 1. Find the domain and range of
1
() 3
fx
x
Solution: Domain of the function is 0x

Range of the function is 1
03
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 0m
.
23
22
(,) (0,0)
lim
xy
x
y
x
y
Solution: (a) Along the line y-axis ( x = 0)
23
22
(,) (0,0)
3
2
(,) (0,0)
3
2
() (0)
() (0)
lim
0
lim 0
lim
lim ( 0
xy
xy
y
y
x
y
x
y
y
y
y
y

  
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Solution of Assignment # 1

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 yx 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: