If $\lim _{x \rightarrow 1} \frac{\sin \left(3 x^{2}-4 x+1\right)-x^{2}+1}{2 x^{3}-7 x^{2}+a x+b}=-2$, then the value of $(a-b)$ is equal to

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Limits

lim_{x \to 1} \frac{s i n ( 3 x ^{2} - 4 x + 1 ) - x ^{2} + 1}{2 x ^{3} - 7 x ^{2} + a x + b} = - 2

, then the value of

(a - b)

is equal to

Answer: 11

For the limit to be finite as

x \to 1

, the denominator must vanish at

x = 1

2 - 7 + a + b = 0

, so

a + b = 5

... (1). Applying L'Hôpital's rule:

lim_{x \to 1} \frac{c o s ( 3 x ^{2} - 4 x + 1 ) ( 6 x - 4 ) - 2 x}{6 x ^{2} - 14 x + a} = - 2

. For this limit to be finite, the denominator must vanish at

x = 1

6 - 14 + a = 0

, so

a = 8

. From (1),

b = - 3

. Therefore,

a - b = 8 - (- 3) = 11

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Integer answer

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