List of top Mathematics Questions on limits of trigonometric functions

\[ \lim_{x \rightarrow \frac{2}{3}} \frac{\sin\left(\pi \cos^2(3x-2)\right)} {9x^2-12x+4} = \ ? \] 

Evaluate: \[ \lim_{x\to \frac{\pi}{2}} \left( \frac{1-\sin x}{\cos x} \right) \]
If \[ \lim_{x\to0}\left(\frac{p\sin2x+1-\cos2x}{x+\tan x}\right)=1, \] then the value of $p$ is:
lim (m→0) (1 - cos m) m² = ____.
lim (m→0) 5sinm m =
The range of the function $y = \log(\sin x)$ where $\sin x > 0$ is:
The value of $\lim_{x \to 0} \dfrac{\sqrt{1 - \cos(x^2)}}{1 - \cos x}$ is equal to:
The value of $\lim_{x \to 0} \dfrac{\sin^2 x}{1 - \cos x}$ is equal to:
Let \( f(x) = \sin x \), \( g(x) = \cos x \), \( h(x) = x^2 \) then
\[ \lim_{x \to 1} \frac{f(g(h(x))) - f(g(h(1)))}{x - 1} = \]
$\lim_{x\rightarrow0}\frac{\sin(\pi \sin^{2}x)}{x^{2}} = $ ________.
The value of $\lim\limits_{x \to 0} \frac{(x-\sin 2x)(2x-\sin x)}{x^2}$ is equal to:
$\lim_{x\rightarrow 2}\frac{\sin x \cos 2 - \cos x \sin 2}{x-2} =$
$\lim_{\theta\rightarrow 0}\frac{\theta \sin 2\theta}{1-\cos 2\theta} =$
$\lim_{x\rightarrow 0}\frac{\sin x}{2\sqrt{2}\sin\frac{x}{\sqrt{2}}} =$
Evaluate the limit \( \lim_{\theta \to \frac{\pi}{2}} \frac{1 - \sin \theta}{\left(\frac{\pi}{2} - \theta\right)\cos \theta} \)
$\displaystyle\lim_{x \to 0}\frac{1 - \cos 4x}{\tan^2 2x}$ is equal to
Find the value of the following expression: \[ \tan^2(\sec^{-1}4) + \cot(\csc^{-1}3) \]
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 \( \lim_{x \to 0} f(x)\).
Given a real-valued function \( f \) such that: \[ f(x) = \begin{cases} \frac{\tan^2\{x\}}{x^2 - \lfloor x \rfloor^2}, & \text{for } x > 0 \\ 1, & \text{for } x = 0 \\ \sqrt{\{x\} \cot\{x\}}, & \text{for } x < 0 \end{cases} \] Then:

Evaluate: $$ \lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\pi - 2x)^3} $$ 

\(\begin{array}{l}\displaystyle \lim_{ x\to 0}\left(\frac{(x+2\cos x)^3 + 2(x+2\cos x)^2 + 3 \sin(x+2\cos x)}{(x+2)^3+2(x+2)^2+3\sin(x+2)}\right)^{\frac{100}{x}} \end{array}\)
is equal to _________.

\(\lim_{{x \to 0}} \limits\) \(\frac{cos(sin x)  -  cos x }{x^4}\) is equal to :

The value of $\lim_{x \to 0} \frac{\cos(mx) - \cos(nx)}{x^2}$ is
\(\lim_{x\to0} \frac{\sin3x - \sin x}{\sin x}\) is
The value of \(\lim_{\theta \to 0} \frac{\tan\theta}{\theta}\) is