lim_(x→0) (((cosx)^(1/m) −(cosx)^(1/n) )/x^2 ) [where m and n integer]


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Question Number 82761 by TANMAY PANACEA last updated on 24/Feb/20

$\lim_{x \to 0} \frac{{(c o s x)}^{\frac{1}{m}} - {(c o s x)}^{\frac{1}{n}}}{x^{2}} [w h e r e m a n d n i n t e g e r]$
Commented by mr W last updated on 24/Feb/20

$\lim_{x \to 0} \frac{{(c o s x)}^{\frac{1}{m}} - {(c o s x)}^{\frac{1}{n}}}{x^{2}}$ $= \lim_{x \to 0} \frac{{(1 - 2 \sin^{2} \frac{x}{2})}^{\frac{1}{m}} - {(1 - 2 \sin^{2} \frac{x}{2})}^{\frac{1}{n}}}{x^{2}}$ $= \lim_{x \to 0} \frac{1 - \frac{2}{m} \sin^{2} \frac{x}{2} + o (\sin^{2} \frac{x}{2}) - [1 - \frac{2}{n} \sin^{2} \frac{x}{2} + o (\sin^{2} \frac{x}{2})]}{x^{2}}$ $= \lim_{x \to 0} \frac{2 (\frac{1}{n} - \frac{1}{m}) \sin^{2} \frac{x}{2}}{x^{2}}$ $= \frac{1}{2} (\frac{1}{n} - \frac{1}{m}) \lim_{x \to 0} {(\frac{\sin \frac{x}{2}}{\frac{x}{2}})}^{2}$ $= \frac{1}{2} (\frac{1}{n} - \frac{1}{m})$
Commented by TANMAY PANACEA last updated on 24/Feb/20

$e x c e l l e n t s i r . . .$
Commented by jagoll last updated on 24/Feb/20

$s i r w h a t i s o (\sin^{2} \frac{x}{2})$
Commented by abdomathmax last updated on 24/Feb/20

$l e t f (x) = \frac{{(c o s x)}^{\frac{1}{m}} - {(c o s x)}^{\frac{1}{n}}}{x^{2}} w e h s v e f o r x \to 0$ $c o s x \sim 1 - \frac{x^{2}}{2} \Rightarrow {(c o s x)}^{\frac{1}{m}} \sim 1 - \frac{x^{2}}{2 m} a n d$ ${(c o s x)}^{\frac{1}{n}} \sim 1 - \frac{x^{2}}{2 n} \Rightarrow {(c o s x)}^{\frac{1}{m}} - {(c o s x)}^{\frac{1}{n}} \sim x^{2} (\frac{1}{2 n} - \frac{1}{2 m})$ $\Rightarrow f (x) \sim \frac{1}{2} (\frac{m - n}{m n}) \Rightarrow {l i m}_{x \to 0} f (x) = \frac{m - n}{2 m n}$
Commented by mr W last updated on 24/Feb/20

$t o j a g o l l s i r :$ $o (x^{2}) s t a n d s f o r t e r m s w i t h h i g h e r o r d e r s$ $t h a n x^{2} . f o r e x a m p l e :$ $\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + . . . . . .$ $= 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + o (x^{4})$ $= 1 - \frac{x^{2}}{2!} + o (x^{2})$ $s e a r c h i n g o o g l e f o r ‘ ‘ l i t t l e o {n o t a t i o n}^{″}$ $f o r m o r e .$
Commented by jagoll last updated on 24/Feb/20

$t h a n k y o u . s i r$
Commented by mathmax by abdo last updated on 24/Feb/20

$t h e s m a l l o (x^{2}) = x^{2} δ (x) w i t h {l i m}_{x \to 0} δ (x) = 0$
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