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I have this homework question where I have to evaluate $\lim_{x\to 0} \frac{\tan(x)−\sin(x)}{x − \sin (x)}$

I realise that when I put in $0$ there is a divison by $0$ problem.

I've tried rewriting the equation by spliting the numerator into two seperate pieces and then evaluate but I'm unsure how I go about this problem.

Cornman
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user135629
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1 Answers1

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Hint:

$$\dfrac{\tan x-\sin x}{x-\sin x}=\dfrac{\sin x(1-\cos x)}{x^3\cos x}\cdot\dfrac{x^3}{x-\sin x}$$

Use Are all limits solvable without L'Hôpital Rule or Series Expansion for the last part.

$$\dfrac{\sin x(1-\cos x)}{x^3\cos x}=\left(\dfrac{\sin x}x\right)^3\cdot\dfrac1{\cos x(1+\cos x)}$$

lab bhattacharjee
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