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

Institute of Mathematical SciencesMathematics

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

roocky
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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

  
docsity.com
pf3

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Download Limits Part 1-Basic Mathematics-Assignment Solution and more Exercises Mathematics in PDF only on Docsity!

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: