CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Continuity & Differentiability

5.1.Continuity.

Q.1. Given that :

∫(1 – cos4x)/x², if x < 0
f(x) = |a if x = 0
∫√x/{√(16 + √x) – 4} if x > 0
If f(x) is continuous at x = 0, find the value of a.

Solution :

For f(x) to be continuous at x = 0, we must have
f(0–) = f(0) = f(0+) ------------------------------ (1)
f(0–) = limx→0– (1 – cos4x)/x²
= limx→0– [(2 sin² 2x)/(x²)]
= limx→0– 8[{(sin²x)/(2x)}²]
= 8.1² = 8. ----------------------- (2)
f(0) = a
f(0+) = limx→0+ [(√x)/{√(16 + √x) – 4}]
= limx→0+ [√x/{√(16 + √x) – 4 }] ×[{ √(16 + √x) + 4}/{√(16 + √x) + 4}]
= limx→0+ [√x{√(16 + √x) + 4]/[(16 + √x) – 16]
= limx→0+ (16 + √x + 4) = 8 ------------------------ (3)
Using (1), (2) and (3) we get a = 8. [Ans.]

Q.2. For what value of k is the following function continuous at x = 2 ?
∫ 2x + 1; x < 2
f(x) = | k ; x = 2
∫ 3x – 1 ; x > 2

Solution :

As f(x) is continuous at x = 2,
limx→2–f(x) = limx→2+f(x) = f(2).
limx→2–(2x + 1) = limx→2+(3x – 1) = k.
Considering limx→2+(3x – 1) = k
Or, 3×2 – 1 = k => k = 5. [Ans.]

Q.3. If f(x), defined by the following, is continuous at x = 0, find the value of a, b and c.
∫sin(a + 1)x + sin x]/x , if x < 0
f(x) = | c if x = 0
∫[√(x + bx²) – √x]/bx3/2 if x > 0.

Solution :

s f(x) is continuous at x = 0, limx→0– f(x) = limx→0+f(x) = f(0)
Therefore, limx→0– f(x) = limx→0– [sin(a + 1)x + sin x]/x
= limx→0– [{sin(a + 1)x}/x + (sin x)/x]
= limx→0– [{sin (a + 1)x}/{(a + 1)x}].(a + 1) + limx→0– [(sinx)/x]
= (a + 1) + 1
= a + 2
= limx→0+ f(x) = limx→0+ [√(x + bx²) – √x]/bx3/2
= limx→0+ [√x{√(1 + bx) – 1}/{bx√x}
= limx→0+ [{√(1 + bx) – 1}/bx]×[{√(1 + bx) + 1}/{√(1 + bx) + 1}]
= limx→0+ [(1 + bx – 1)/{bx(√(1 + bx) + 1)}]
= limx→0+ [1/{√(1 + bx) + 1}]
= 1/(√1 + 1) = 1/2
Therefore, a + 2 = 1/2 = c => a = – 3/2, c = 1/2. [Ans.]

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289