Q.6. The largest value of the non-negative integer a for which
lim x 1 - a x + sin x - 1 + a x + sin x - 1 - 1 1 - x 1 - x = 1 4 is

Dear student
limx1-ax+sin(x-1)+ax+sin(x-1)-11-x1-x=14limx1sin(x-1)+a(1-x)(x-1)+sin(x-1)1-x1+x1-x=14limx1sin(x-1)x-1-a1+sin(x-1)x-11+x=14Let f(x)=limx1sin(x-1)x-1-a1+sin(x-1)x-1 and g(x)=limx11+xSo, considerlimx1sin(x-1)x-1-a1+sin(x-1)x-1=limx1sin(x-1)x-1-alimx11+sin(x-1)x-1limx1sin(x-1)x-1-limx1alimx11+limx1sin(x-1)x-1Consider,limx1sin(x-1)x-1Apply L'hopital's rule ,we getlimx1cos(x-1)1=cos0=1So,limx1sin(x-1)x-1-limx1alimx11+limx1sin(x-1)x-1=1-a1+1=1-a2and consider,g(x)=limx11+x=2So,limx1sin(x-1)x-1-a1+sin(x-1)x-11+x=141-a22=14a-12=1a2-2a+1=1a2-2a=0a(a-2)=0a=0 or a=2But for a=2 base of above limit approaches -12 and exponent approachesto 2 and since base cannot be negative hence limit does not exist.
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