Let α and β be the distinct roots of ax2 + bx + c = 0 , then displaystyle limx →α (1- cos (ax2 + bx + c)/(x-α)2) is equal to

Question

Let α and β be the distinct roots of ax2 + bx + c = 0 , then displaystyle limx →α (1- cos (ax2 + bx + c)/(x-α)2) is equal to (A) (a2/2) (α - β)2 (B) 0 (C) (-a2/2) (α - β )2 (D) (1/2) (α - β )2

Tardigrade Admin · Accepted Answer

Given limit = displaystyle limx →α (1- cos a(x -α)(x-β)/(x -α)2) = displaystyle limx →α (2 sin2 (a ((x -α)(x - β)/2))/(x -α)2) = displaystyle limx →α (2/(x -α)2) ×( sin2(a ((x-α)(x -β)/2))/(a2(x-α)2(x-β)2)4) ×(a2 (x-α)2 (x -β)2/4) = (a2(α - β)2/2) .

Q. Let $α$ and $β$ be the distinct roots of $a x^{2} + b x + c = 0$ , then $x \to α lim \frac{1 - cos ( a x ^{2} + b x + c )}{( x - α ) ^{2}}$ is equal to

$\frac{a ^{2}}{2} (α - β)^{2}$

0

$\frac{- a ^{2}}{2} (α - β)^{2}$

$\frac{1}{2} (α - β)^{2}$

Q. Let α and β be the distinct roots of ax2+bx+c=0 , then x→αlim​(x−α)21−cos(ax2+bx+c)​ is equal to

2a2​(α−β)2

0

2−a2​(α−β)2

21​(α−β)2

Given limit =x→αlim​(x−α)21−cosa(x−α)(x−β)​ =x→αlim​(x−α)22sin2(a2(x−α)(x−β)​)​ =x→αlim​(x−α)22​×4a2(x−α)2(x−β)2​sin2(a2(x−α)(x−β)​)​×4a2(x−α)2(x−β)2​ =2a2(α−β)2​.

Q. Let $α$ and $β$ be the distinct roots of $a x^{2} + b x + c = 0$ , then $x \to α lim \frac{1 - cos ( a x ^{2} + b x + c )}{( x - α ) ^{2}}$ is equal to

$\frac{a ^{2}}{2} (α - β)^{2}$

$\frac{- a ^{2}}{2} (α - β)^{2}$

$\frac{1}{2} (α - β)^{2}$