New Exam Prep

$\lim_{x \to 0} \frac{(27+x)^{1/3}-3}{9-(27+x)^{2/3}}$ equal to $k$, then $|12k| = $ _________

The full exam-length answer is in the app.

The value of $\lim_{x \to \infty} \frac{x}{x + \frac{x}{\sqrt[3]{x} + \frac{x}{\sqrt[3]{x} + \dots \infty}}}$ is _________

The full exam-length answer is in the app.

$\lim_{x \to \infty} \frac{3\left[\frac{x}{4}\right]}{x} = \frac{p}{q}$ (where [.] denotes greatest integer function), then $p+q = $ _________ (where $p,q$ are relative prime) is _________

The full exam-length answer is in the app.

$\lim_{x \to \frac{\pi}{2}} \frac{\sin\left(\frac{\pi}{2}-x\right)}{2\cos x - 1}$ is equal to $k$, then $4\sqrt{3}k = $ _________

The full exam-length answer is in the app.

If the domain of the function $f(x) = \cos^{-1}\left(\frac{2-|x|}{4}\right) + \{\log_e(3-x)\}^{-1}$ is $[-\alpha, \beta) - \{\gamma\}$, then the value of $\alpha+\beta+\gamma$ is equal to: _________

The full exam-length answer is in the app.

$\lim_{n \to \infty} \tan^{-1}(2) \left( \frac{n}{n^2+1} + \frac{n}{n^2+2^2} + \dots + \frac{n}{n^2+n^2} \right)$

The full exam-length answer is in the app.

$\lim_{x \to 0^+} \frac{-1 + \sqrt{(\tan x - \sin x) + \sqrt{(\tan x - \sin x) + \sqrt{\tan x - \sin x + \dots \infty}}}}{-1 + \sqrt{x^3 + \sqrt{x^3 + \sqrt{x^3 + \dots \infty}}}}$

ABC is an isosceles triangle inscribed in a circle of radius r. If $AB = AC$ and $h$ is the altitude from A to BC, then the triangle ABC has perimeter $P = 2\left[\sqrt{(2hr-h^2)} + \sqrt{2hr}\right]$ and area A, then $\lim_{h \to 0} \frac{A}{P^3}$

The full exam-length answer is in the app.

$\lim_{x \to 0} \frac{\sin(\pi \cos^2(\tan(\sin x)))}{x^2}$

The full exam-length answer is in the app.

$\lim_{n \to \infty} \sum_{r=1}^n \cot^{-1}\left(\frac{r^4+r^2+2}{2r}\right) = $

The full exam-length answer is in the app.

Let $a = \lim_{x \to 0} \frac{\ln(\cos 2x)}{3x^2}$, $b = \lim_{x \to 0} \frac{\sin^2 2x}{x(1-e^x)}$, $c = \lim_{x \to 1} \frac{\sqrt{x-x}}{\ln x}$. Then $a,b,c$ satisfy

The full exam-length answer is in the app.