Evaluate the following limits: lim x → 0 cos a x − cos b x cos c x − 1 - Mathematics

Question

Evaluate the following limits:

$lim_{x \to 0} \frac{\cos a x - \cos b x}{\cos c x - 1}$

Solution

$lim_{x \to 0} \frac{\cos a x - \cos b x}{\cos c x - 1}$

$= lim_{x \to 0} \frac{- 2 \sin (\frac{a x + b x}{2}) \sin (\frac{a x - b x}{2})}{- 2 \sin^{2} \frac{c x}{2}} (1 - \cos 2 θ = 2 \sin^{2} θ)$

$= lim_{x \to 0} \frac{(\frac{a + b}{2}) x \times \frac{\sin (\frac{a + b}{2}) x}{(\frac{a + b}{2}) x} \times (\frac{a - b}{2}) x \times \frac{\sin (\frac{a - b}{2}) x}{(\frac{a - b}{2}) x}}{\frac{c^{2} x^{2}}{4} \times \frac{\sin^{2} \frac{c x}{2}}{\frac{c^{2} x^{2}}{4}}}$

$= \frac{(a + b) (a - b)}{c^{2}} \times \frac{lim_{x \to 0} \frac{\sin (\frac{a + b}{2}) x}{(\frac{a + b}{2}) x} \times lim_{x \to 0} \frac{\sin (\frac{a - b}{2}) x}{(\frac{a - b}{2}) x}}{{(lim_{x \to 0} \frac{\sin \frac{c x}{2}}{\frac{c x}{2}})}^{2}}$

$= \frac{a^{2} - b^{2}}{c^{2}} \times \frac{1 \times 1}{1} (lim_{θ} \to 0 \frac{\sin θ}{θ} = 1)$

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

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Concept of Limits - Algebra of Limits

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Chapter 29: Limits - Exercise 29.7 [Page 51]

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Chapter 29 Limits
Exercise 29.7 | Q 61 | Page 51

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Evaluate the following limits: lim x → 0 cos a x − cos b x cos c x − 1 - Mathematics

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