displaystyle limx → 0 (cos ax-cos bx/x2) is equal to

Question

displaystyle limx → 0 (cos ax-cos bx/x2) is equal to (A)  (a2-b2/2)  (B)  (b2-a2/2)  (C)  a2 - b2  (D)  b2 - a2

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 ( cos a x- cos b x/x2) = displaystyle lim x arrow 0 (-a sin a x+b sin b x/2 x) (using L'Hospital's rule) = displaystyle lim x arrow 0 (-a2 cos a x+b2 sin b x/2) (using L'Hospital's rule) =(b2-a2/2)

Q. $x \to 0 lim$ $\frac{cos a x - cos b x}{x ^{2}}$ is equal to

$\frac{a ^{2} - b ^{2}}{2}$

$\frac{b ^{2} - a ^{2}}{2}$

$a^{2} - b^{2}$

$b^{2} - a^{2}$

$x \to 0 lim \frac{cos a x - cos b x}{x ^{2}}$
$= x \to 0 lim \frac{- a sin a x + b sin b x}{2 x}$
(using L'Hospital's rule)
$= x \to 0 lim \frac{- a ^{2} cos a x + b ^{2} sin b x}{2}$
(using L'Hospital's rule)
$= \frac{b ^{2} - a ^{2}}{2}$

Q. x→0lim​ x2cosax−cosbx​ is equal to

2a2−b2​

2b2−a2​

a2−b2

b2−a2