Let α and β be the distinct roots of a x2+b x+c=0, then displaystyle lim x arrow α (1- cos (a x2+b x+c)/(x-α)2) is equal to

Question

Let α and β be the distinct roots of a x2+b x+c=0, then displaystyle lim x arrow α (1- cos (a x2+b x+c)/(x-α)2) is equal to (A) (1/2)(α-β)2 (B) (-a2/2)(α-β)2 (C) 0 (D) (a2/2)(α-β)2

Tardigrade Admin · Accepted Answer

Now, displaystyle lim x arrow α (1- cos (a x2+b x+c)/(x-α)2) = displaystyle lim x arrow α (2 sin 2((a x2+b x+c/2))/(x-α)2) = displaystyle lim x arrow α (2 sin 2((a/2)(x-α)(x-β))/((a)2)2(x-α)2(x-β)2)((a/2))2(x-β)2 = displaystyle lim x arrow α (a2/2)(x-β)2[∵ lim x arrow 0 ( sin x/x)=1] =(a2/2)(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

$\frac{1}{2} (α - β)^{2}$

$\frac{- a ^{2}}{2} (α - β)^{2}$

0

$\frac{a ^{2}}{2} (α - β)^{2}$

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

21​(α−β)2

2−a2​(α−β)2

0

2a2​(α−β)2

Now, x→αlim​(x−α)21−cos(ax2+bx+c)​ =x→αlim​(x−α)22sin2(2ax2+bx+c​)​ =x→αlim​(2a​)2(x−α)2(x−β)22sin2(2a​(x−α)(x−β))​(2a​)2(x−β)2 =x→αlim​2a2​(x−β)2[∵x→0lim​xsinx​=1] =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

$\frac{1}{2} (α - β)^{2}$

$\frac{- a ^{2}}{2} (α - β)^{2}$

$\frac{a ^{2}}{2} (α - β)^{2}$