EXERCISE - A
Find the point/s of Discontinuty:
1) (x²-4)/(x+2). -2
2) (x²-1)/(x-1). 1
3) f(x)= (x²+3x-1)/(x²-7x+12). 3,4
4) f(x)= (2x²-5x+2)/(x²-3x+2). 1,2
5) f(x)= |x²-3x|/(x-3) when x≠3
5. When x= 3. 3
6) f(x)= x²-9 when x≤3
8. When x= 3. 3
7) f(x)= tanx/x + sinx when x≠0
Cosx + 2 when x= 0. 0
EXERCISE - B
Discuss the continuity:::
1)a) f(x)= (x²-9)/(x-3) at x= 4. Y
b) f(x)= (x²-25)/(x-5) when x≠5
10. When x=5. Y
c) f(x)= (x²-25)/(x-5) when x≠5
8. When x=5.
At x= 5. N
d) f(x)= x if x ≥ 0
= 1 if x< 0 at origin. N
e) f(x)= (x²-4x+3)/(x²-1) ; x ≠1
2 ; x = 1
at x= 1. N
f) f(x)= (1- xⁿ)/(1- x); x ≠ 1
n -1, x= 1
At x= 1, n ∈ N. N
2)a) f(x)= (3-x²)/2 when x<2
4-x² When x≥2 N
At x= 2
b) f(x)= 1+x ; x ≤ 2
= 5 - x ; x > 2 at x= 2. Y
c) f(x)= x -1 ; 1≤x ≤ 2
2x -3 ; 2 ≤x≤3 at x=2. Y
d) f(x)= 3x-2 ; x ≤ 0
1 + x ; x > 0 at x= 0. N
e) f(x)= 5x-4 ; 0< x ≤ 1
4x³-3x ; 1<x≤ 2 at x= 1. Y
f) f(x)= 2x -1, x< 0
2x+1, x≥ 0 at x = 0. N
3a) f(x)= (x²-16)/(x-4) when x<4
8 When x=4.
(3x+4)/2 when x>4
At x= 4. Y
b) x+ 2 ; x < 1
f(x)= 0 ; x= 1
x - 2; x> 1. N
c) f(x)= 2x²-1. when 0<x<2
4x-1 When 2≤x<3
x+4 when x≥ 3
At x= 2
d) x , x> 0
f(x)= 1, x=0 at x= 0 N
- x, x< 0
e) x, 0≤ x < 1/2
f(x)= 1/2, x= 1/2 at x= 1/2 y
1- x, 1/2< x < 1
4a) f(x)= cosx, if x≥0
- cosx if x< 0 at x= 0. N
b) f(x)= tan(2x)/3x ; x ≠0
2/3 ; x =0 at x= 0. Y
c) f(x)= (x-a) cos{1/(x-a)} ; x ≠a
0 ; x= a
At x= a. Y
d) f(x)= (1-cosx)/x² ; x≠0
1 ; x = 0
At x= 0. N
e) f(x)= x² sin(1/x); x≠ 0
0; x= 0
At x= 0. Y
f) f(x)= (x -a) sin{1/(x -a)}; x≠ a
0, x= a
At x= a. Y
g) f(x)= (eˣ -1)/log(1+ 2x), if x≠ 0
7, if x= 0
At x= 0. N
5a) f(x)= x/|x|, x≠ 0
1, x= 0. N
b) f(x)= (x - |x|)/x when x≠ 0
2 when x< 0, at x= 0.
c) f(x)= |x²- 1|/(x -1), for x ≠ 1
2, for x=1
At x= 1. N
d) f(x)= {2|x|+ x²}/x; x ≠ 0
0, x= 0
At x= 0. N
e) f(x)= |x| cos(1/x); x ≠ 0
0, x=0
At x= 0. N
EXERCISE - C
PROVE For Discontinuous
1a) f(x)= x² when x≠ 1
2 when x= 1 at x= 1
b) f(x)= x² ; x ≤ 1
5 + x ; x > 1 at x= 1.
c) f(x)= x² ; 1 ≤ x< 2
3x-4 ; 2≤ x< 4 at x= 2
2) f(x)= 1+ x², if 0≤ x < 1
2 - x, if x > 1 at x = 1
3) x, when 0 ≤ x < 1/2
f(x)= 1 when x= 1/2
1- x when 1/2<x<1
4a) f(x)= (1- cosx)/x²; when x ≠ 0
1, when x= 0
At x= 0
b) f(x)= (sin2x)/x when x≠0
1 when x=0
At x= 0
5a) f(x)= {x - |x|}/2; when x ≠ 0
2; when x=0
At x= 0
b) f(x)= |x - a|/(x - a), when x ≠ a
1, when x = a
At x= a
EXERCISE - D
prove for continuous:
1a) f(x)= (x²- x -6)/(x -3); if x≠ 3
5, if x= 3
At x=3.
b) f(x)= (x²-9)/(x -3); if x ≠3
6; if x=3
At x= 3
c) f(x)= (x² - 1)/(x -1); if x≠1
2 if x= 1
at x= 1
2) f(x)= x² ; 1 ≤ x<2
= 3x-4 ; 2≤ x<4
At x= 3.
3a) If f(x)= (sin3x)/x; when x ≠ 0
1; when x= 0
At x= 0
b) f(x)= (sinx)/x + cosx if x≠0
2. if x= 0 at x= 0
c) (sin3x)/(tan2x), if x< 0
f(x)= 3/2 , if x= 0
Log(1+ 3x)/(e²ˣ-1), if x > 0
At x= 0
EXERCISE - E
Find the value of K. If function is continuous.
1a) f(x)= (x²-16)/(x-4) If x≠ 4
K If x= 4. 8
At x= 4
b) f(x)= (x²-1)/(x -1), x≠ 1
K, x= 1. 2
At x= 1
c) f(x)= (x²- 3x+2)/(x -1), if x≠ 1
K, if x =1. -1
At x= 1
d) f(x)= kx², if x ≤ 2
3, if x> 2 at x=2. 3/4
e) f(x)= kx², x ≥ 1
4, x < 1 at x= 1. 4
2a) Kx+ 5 ; x≤2
f(x)= x -1 ; x > 2 At x= 2. -2
b) f(x)= ax +5; if x≤ 2
x - 1; if x> 2 at x=2. -2
c) f(x)= Kx +1, if x ≤ 5
(3x -5), if x > 5
At x= 5. 9/5
d) (x²- 25)/(x -5), x≠ 5
K, x= 5 at x= 5. 0
e) K(x²+2) ; x≤0
f(x)= 3x +1 ; x>0
Continuous at x= 0? Also, Write whether the function is continuous at x= 1. 1/2, yes
3a) 3ax+ b ; x >1
f(x)= 11 ;. x= 1
5ax - 2b; x< 1 is continuous at x=1, find a, b. 3,2
b) x+ 2 ; x ≤ 2
f(x)= ax + b ; 2< x<5
3x - 2 ; x≥5
The function is continuous, find a, b. 3, -2
c) 1, if x≤ 3
f(x)= ax+ b, if 3< x< 5
7, if x ≥ 5
At x= 3 and x = 5, find a, b. 3,-8
4a) {sin(a+1)x + sinx}/x, for x< 0
f(x)= c for x= 0
{√(x + bx²) - √x}/bx³⁾² for x> 0
At x= 0, find a,b,c. -3/3, b∈R - {0}, 1/2
b) f(x)= (sin5x)/3x, if x ≠ 0
K, if x = 0.
At x= 0. 5/3
c) f(x)= (sin2x)/5x, if x≠ 0
K, if x= 0
At x= 0. 2/5
d) f(x)= (sin2x)/x, x≠ 0
K, x= 0 at x= 0. 2
e) f(x)= (k cosx)/(π- 2x), x=π/2
3, x= π/2
At x= π/2. 6
f) f(x)= (1- Kx)/(x sinx), x≠ 0
1/2, x= 0
At x= 0. ±1
g) f(x)= (1- cos4x)/8x², when x≠ 0
K, when x= 0 at x= 0. 1
h) f(x)= (1- cos2x)/x², if x≠ 0
8, if x= 0 at x= 0. ±2
i) f(x)= (x -1) tan(πx/2), if x≠ 1
K, if x= 1. At x=1. -2/π
j) f(x)= k(x²- 2x), if x < 0
Cosx, if x≥ 0 at x=0. No value
j) If f(x)= (cos²x - sin²x -1)/{√(x½+1) -1}, x≠ 0
K, x= 0 at x= 0. -4
5) f(x)= (2ˣ⁺² - 16)/(4ˣ-16), if x≠ 2
K, x=2 at x=2. 1/2
6) {(x -4)/|x -4|} + a, if x< 4
f(x)= a+ b, if x= 4
{(x -4)/|x -4|} + b, x > 4
At x= 4, find ab. 1.-1
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