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PleaPl help me with the question given in the image attachment.

Anshu Mishra , 5 Years ago

Grade 12th pass

2 Answers

Arun

Last Activity: 5 Years ago

Lim x->a (cos x - cos a / cot x - cot a)
= Lim x->a (cos x - cos a) / [(cos x sin a - cos a sin x)/(sin x sin a)]
= Lim x->a (-2sin((x-a)/2)sin((x+a)/2)) / [(sin(a-x))/(sin x sin a)]
= sin^3 a
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Ideas: cos x - cos a = cos[(x+a)/2 + (x-a)/2] - cos[(x+a)/2 - (x-a)/2] = -2sin((x-a)/2)sin((x+a)/2)

Samyak Jain

Last Activity: 5 Years ago

lim (cosx – cosa) / (cotx – cota) It is of 0 by 0 form. Simplifying it we can calculate the limit.

x $\rightarrow$ a

We know cosx – cosa = 2 sin(x+a)/2 sin(a – x)/2 and

cotx – cota = cosx/sinx – cosa/sina = (sina cosx – cosa sinx) / sinx sina = sin(a – x) / sinx sina

$\therefore$ lim (cosx – cosa) / (cotx – cota) = lim [2 sin(x+a)/2 sin(a – x)/2] / [sin(a – x) / sinx sina]

x $\rightarrow$ a x $\rightarrow$ a

= lim [2 sin(x+a)/2 sin(a – x)/2 . sinx sina ] / [2 sin (a – x)/2 cos(a – x)/2 ]

x $\rightarrow$ a

= lim sin(x+a)/2 . sinx sina / cos(a – x)/2 = sina lim sin(x+a)/2 . sinx / lim cos(a – x)/2

x $\rightarrow$ a x $\rightarrow$ a x $\rightarrow$ a

= sina sin(a+a)/2 . sina / cos(a – a)/2 = sina sina .sina / cos0

= sin³a

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