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Evaluate the following limit: limx→0sin⁡(α+β)x+sin⁡(α−β)x+sin⁡2αxcos2⁡βx−cos2⁡αx

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Question

Evaluate the following limit: 

\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]

Sum
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Solution

\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]

`therefore cos^2 beta x - cos^2alpha x`

`= cos^2betax - cos^2alphax + cos^2betax * cos^2alphax - cos^2betax * cos^2alphax`

= `cos^2 beta x - cos^2 beta x * cos^2 alpha x - cos^2 alpha x + cos^2 beta x * cos^2 alpha x`

= `cos^2 beta x (1 - cos^2 alpha x) - cos^2 alpha x (1 - cos^2 beta x)`

= `cos^2 beta x * sin^2 alpha x - cos^2 alpha x * sin^2 beta x`

= `(cos beta x * sin alpha x)^2 - (cos alpha x * sin beta x)`

= `(cos beta x * sin alpha x + cos alpha x * sin beta x)(cos beta x * sin alpha x - cos alpha x * sin beta x)`

= `sin (alpha x + beta x)  sin (alphax - betax)`    ...[sin (A ± B) = sin A cos B ± cos A sin B]

= `sin (alpha + beta)x  sin (alpha - beta)x`

`therefore lim_(x->0) (x{sin (alpha + beta )x + sin (alpha - beta)x + sin 2 alpha x})/(sin (alpha + beta)x  sin (alpha - beta) x)`

`lim_(x->0) ((x{sin (alpha + beta )x + sin (alpha - beta)x + sin 2 alpha x})/x^2)/((sin (alpha + beta)x  sin (alpha - beta) x)/x^2)`

`lim_(x->0) ((sin (alpha + beta)x)/x + (sin (alpha - beta)x)/x + (sin 2 alpha x)/x)/((sin (alpha + beta)x)/x xx (sin (alpha - beta)x)/x)`

`lim_(x->0) ((sin (alpha + beta)x)/((alpha + beta) x) xx (alpha + beta) + (sin (alpha - beta)x)/((alpha - beta)x) xx (alpha - beta) + (sin 2 alpha x)/(2alphax) xx 2alpha)/((sin (alpha + beta)x)/((alpha + beta)x) xx (alpha + beta) xx (sin (alpha - beta)x)/((alpha - beta)x) xx (alpha - beta))`

`= (alpha + beta + alpha - beta + 2alpha)/((alpha + beta)(alpha - beta))`

`= (4alpha)/(alpha^2 - beta^2)`

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Chapter 29: Limits - Exercise 29.7 [Page 51]

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R.D. Sharma Mathematics [English] Class 11
Chapter 29 Limits
Exercise 29.7 | Q 60 | Page 51

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