List of top Mathematics Questions on limits of trigonometric functions asked in MET

Let \( f(x) = \left[\frac{\sin x}{x}\right] + \left[\frac{2\sin x}{x}\right] + \cdots + \left[\frac{10\sin x}{x}\right] \) (where \([\,]\) is the greatest integer function). Find \( \lim_{x \to 0} f(x)\).

\(\lim_{x\to0} \frac{\sin3x - \sin x}{\sin x}\) is

The value of \(\lim_{\theta \to 0} \frac{\tan\theta}{\theta}\) is

The value of \( \lim_{x \to 0} \left( \frac{a^x + b^x + c^x}{3} \right)^{\frac{2}{x}} \), \( (a,b,c>0) \) is

\( \lim_{x \to \frac{\pi}{2}} \frac{(1 - \tan \frac{x}{2})(1 - \sin x)}{(1 + \tan \frac{x}{2})(\pi - 2x)^3} \) is equal to

The value of \( \lim_{x \to 0} \frac{1 - \cos x}{x^{2}} \) is: