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 Share with your friends Share 3 Lovina Kansal answered this Dear student limx→1-ax+sin(x-1)+ax+sin(x-1)-11-x1-x=14⇒limx→1sin(x-1)+a(1-x)(x-1)+sin(x-1)1-x1+x1-x=14⇒limx→1sin(x-1)x-1-a1+sin(x-1)x-11+x=14Let f(x)=limx→1sin(x-1)x-1-a1+sin(x-1)x-1 and g(x)=limx→11+xSo, considerlimx→1sin(x-1)x-1-a1+sin(x-1)x-1=limx→1sin(x-1)x-1-alimx→11+sin(x-1)x-1limx→1sin(x-1)x-1-limx→1alimx→11+limx→1sin(x-1)x-1Consider,limx→1sin(x-1)x-1Apply L'hopital's rule ,we getlimx→1cos(x-1)1=cos0=1So,limx→1sin(x-1)x-1-limx→1alimx→11+limx→1sin(x-1)x-1=1-a1+1=1-a2and consider,g(x)=limx→11+x=2So,limx→1sin(x-1)x-1-a1+sin(x-1)x-11+x=14⇒1-a22=14⇒a-12=1⇒a2-2a+1=1⇒a2-2a=0⇒a(a-2)=0⇒a=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. Regards 1 View Full Answer