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  • If <img alt="\lim _{x \rightarrow 1} \frac{a \sin (x-1)+b \cos (x-1)+4}{x^2-1}=-2" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%201%7D%20%5Cfra

If \lim _{x \rightarrow 1} \frac{a \sin (x-1)+b \cos (x-1)+4}{x^2-1}=-2 , then (a,b) is equal to ............

 

Option: 1

(-2,1)


Option: 2

(0,1)


Option: 3

(0,2)

 


Option: 4

(-4,-4)


Answers (1)

best_answer

\therefore RHS is finite quantity

\therefore At x\rightarrow 1, Numerator must be = 0 

                        \begin{gathered} 0+b+4=0 \\ \\b=-4 \end{gathered}

Then           \lim _{x \rightarrow 1} \frac{a \sin (x-1)-4 \cos (x-1)+4}{\left(x^2-1\right)}=-2

Put                             x=1+h

Then           \lim _{h \rightarrow 0} \frac{a \sin h+4(1-\cos h)}{h(2+h)}=-2

\begin{array}{ll} \Rightarrow & \operatorname{Lim}_{h \rightarrow 0} \frac{a\left(\frac{\sin h}{h}\right)+4\left(\frac{1-\cos h}{h}\right)}{2+h}=-2 \\ \\\Rightarrow & \frac{a(1)+0}{2}=-2 \\ \\\therefore \: \: a = -4\\ \\\Rightarrow & (a, b)=(-4,-4) \end{array}

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Rishabh

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