List of top Mathematics Questions on Limits asked in COMEDK UGET

Let $ \alpha $ and $ \beta $ be the distinct roots of $ ax^2 + bx + c = 0 $, then $ \lim_{x \to \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2} $ is equal to:

The function defined by \[ f(x) = \begin{cases} \frac{\sin x}{x} + \cos x & \text{if } x>0 \\ -5k & \text{if } x = 0 \\ \frac{4(1 - \sqrt{1 - x})}{x} & \text{if } x<0 \end{cases} \] is continuous at

\[ \lim_{x \to 0} \frac{a^x - b^x}{x} \text{ is equal to:} \]

The value of $ \lim_{x \to \infty} \left( \frac{x^2 - 2x + 1}{x^2 - 4x + 2} \right)^{2x} \text{ is} $

The value of $ \lim_{x \to 0} \frac{e^{ax} - e^{bx}}{2x} \text{ is equal to} $

If \( \lim_{x \to 0} \frac{(1 + a^3)^x + 8e^{1/x}}{1 + (1 - b^3)^x e^{1/x}} = 2 \), then:

The approximate value of \( (0.007)^{1/3} \) is:

If \( \lim_{x \to \infty} \left( 1 + \frac{a}{x} + \frac{b}{x^2} \right)^{2x} = e^2 \), then:

The argument of \( \dfrac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \) is:

\( (i + \sqrt{3})^{100} + (i - \sqrt{3})^{100} + 2^{100} \) is equal to:

Evaluate the integral \( \int \frac{1}{x \sqrt{ax^2 - x^2}} \, dx \)

Evaluate the integral \( 3 \int \frac{3^x}{\sqrt{1 - 9^x}} \, dx \)

Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx \)

The solution of the differential equation \( \frac{d^2y}{dx^2} = 0 \) represents

The solution of the differential equation \( \frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0 \) is

The solution of the differential equation \( x \frac{dy}{dx} = \cot y \) is

If \( \sin A + \sin B = a \) and \( \cos A + \cos B = b \), then \( \cos(A + B) \) equals?

What is \( \frac{\cos \theta}{1 - \tan \theta} + \frac{\sin \theta}{1 - \cot \theta} \) equal to?

Find the general solution of \( \sin 2x + \cos x = 0 \).

If \( f(x) + 2f(1 - x) = x^2 + 5 \) for all real values of \( x \), then \( f(x) \) is given by:

Evaluate $\left[i^{22} + \left(\dfrac{1}{i}\right)^{25}\right]^3$

The value of \( 3\log_5{5} - 5\log_4{3} \) is