If α, β are the distinct roots of x 2+ bx + c =0 then displaystyle lim x arrow β (e2(x2+b x+c)-1-2(x2+b x+c)/(x-β)2) is equal to:

Question

If α, β are the distinct roots of x 2+ bx + c =0 then displaystyle lim x arrow β (e2(x2+b x+c)-1-2(x2+b x+c)/(x-β)2) is equal to: (A) b2+4 c (B) 2(b2+4 c) (C) 2(b2-4 c) (D) b2-4 c

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow β (e2(x2+b x+c)-1-2(x2+b x+c)/(x-β)2) ⇒ displaystyle lim x arrow β (1(1+(2(x2+b x+c)/1 !)+(22(x2+b x+c)2/2 !)+ ldots)-1-2(x2+b x+c)/(x-β)2) ⇒ displaystyle lim x arrow β (2(x2+b x+1)2/(x-β)2) ⇒ displaystyle lim x arrow β (2(x-α)2(x-β)2/(x-β)2) ⇒ 2(β-α)2=2(b2-4 c)

Q. If $α, β$ are the distinct roots of $x^{2} + b x + c = 0$ then $x \to β lim \frac{e ^{2 (x^{2} + b x + c)} - 1 - 2 ( x ^{2} + b x + c )}{( x - β ) ^{2}}$ is equal to:

$b^{2} + 4 c$

$2 (b^{2} + 4 c)$

$2 (b^{2} - 4 c)$

$b^{2} - 4 c$

Q. If α,β are the distinct roots of x2+bx+c=0 then x→βlim​(x−β)2e2(x2+bx+c)−1−2(x2+bx+c)​ is equal to:

b2+4c

2(b2+4c)

2(b2−4c)

b2−4c

x→βlim​(x−β)2e2(x2+bx+c)−1−2(x2+bx+c)​ ⇒x→βlim​(x−β)21(1+1!2(x2+bx+c)​+2!22(x2+bx+c)2​+…)−1−2(x2+bx+c)​ ⇒x→βlim​(x−β)22(x2+bx+1)2​ ⇒x→βlim​(x−β)22(x−α)2(x−β)2​ ⇒2(β−α)2=2(b2−4c)

Q. If $α, β$ are the distinct roots of $x^{2} + b x + c = 0$ then $x \to β lim \frac{e ^{2 (x^{2} + b x + c)} - 1 - 2 ( x ^{2} + b x + c )}{( x - β ) ^{2}}$ is equal to:

$b^{2} + 4 c$

$2 (b^{2} + 4 c)$

$2 (b^{2} - 4 c)$

$b^{2} - 4 c$