If _ x 1^ + (x-1)(6+ (x-1))+ (1-x) (x-1)^ 3 =-1, where , R, then … - Vidyadip

MCQ

If $\lim _{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^{3}}=-1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda+\mu$ is equal to

✓
18
B
20
C
19
D
17

✓

Answer

Correct option: A.

18

(A) 18
Put $\mathrm{x}=1+\mathrm{h}$
$\lim _{h \rightarrow 0} \frac{h(6+\lambda \cosh )-\mu \sinh }{h^{3}}=-1$
$\lim _{h \rightarrow 0} \frac{h\left(6+\lambda\left(1-\frac{h^{2}}{2!}\right)\right)-\mu\left(h-\frac{h^{3}}{3!}\right)}{h^{3}}=-1$
$6+\lambda-\mu=0$ and $-\frac{\lambda}{2}+\frac{\mu}{6}=-1$
$\lambda+\mu=18$

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