Limits Questions - JEE

If a is the positive root of the equation p(x) = x2 − x − 2 = 0, then limx → a 1 − cos(πp(x))/x2 + a − 4 is equal to

The value of $\lim_{x \to \pi/4} (1 + [x])^{\log(\tan x)}$ (where $[\cdot]$ denotes greatest integer function) is

If $\lim_{x \to 0} \frac{1}{x^8}\left[\cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2} - \cos\frac{x^2}{4}\right] = 2k$, then the value of $k$ is ……… .

Find $\lim_{x \to 0} \frac{\log \log (1 - x^2)}{\log \log \cos x}$

If f(x) = cos x × cos 2x × cos 4x × cos 8x × cos 16x, then f'(π/4) is

If y = sin⁻¹((1-x)/(1+x)), 0 , then dy/dx is

The points of discontinuity of $y = \frac{1}{u^2 + u - 2}$, where $u = \frac{1}{x-1}$, are:

Given $f(0) = 0$ and $f(x) = \frac{1}{1-e^{-1/x}}$ for $x \neq 0$. Then, only one of the following statements on $f(x)$ is true.

If x\log_e(\log_e x) - x^2 + y^2 = 4 (y > 0), then \frac{dy}{dx} at x = e is equal to

If $f(x) = \cos\left[\frac{\pi}{x}\right]\cos\left(\frac{\pi(x-1)}{2}\right)$, where $[\cdot]$ denotes the greatest integer function, then $f(x)$ is continuous at

Let $(\tan\alpha)x + (\sin\alpha)y = \alpha$ and $(\alpha\operatorname{cosec}\alpha)x + \cos\alpha y = 1$ be two variable straight lines, $\alpha$ being the parameter. Let P be the point of intersection of the lines. If the coordinates of P in the limiting position when $\alpha \to 0$ be $(h, k)$ then

Given,\[f(x) = \begin{cases} \frac{\cos 2x - \sin 2x - 1}{x^2 + x - 2}, & x \neq \text{?} \\ a, & x = \text{?} \end{cases}\]where f is continuous at the point where the function is defined piecewise. Find the constant a.

If f'(x) = g(x) and g'(x) = -f(x) for all x and f(2) = 4 = f'(2), then (f(24))^2 + (g(24))^2 is

If the function f(x) = {a|p - x| + 1, x ≤ 5; b|x - p| + 3, x > 5} is continuous at x = 5, then the value of a - b is

If $y = \tan^{-1}(\sec x - \tan x)$, then $\frac{dy}{dx}$ is equal to

If $\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{(x-a)^3}$ is a finite non-zero number, then $f(x)$ is of maximum degree

The value of $\lim_{x \to 0} \sin^{-1}\{x\}$ (where $\{\cdot\}$ denotes fractional part of $x$) is

$\lim_{x \to \infty} C_n^m x^m \left(1 + \frac{1}{n}\right)^{nx} \left(1 + \frac{1}{x}\right)^m$ equals to

Let $h(x) = \min\{x; x^2\}$ for every real number $x$. Then, which one of the following is true?

The value of $\lim_{x \to 0} \frac{\tan(\{1/x\}) \sin(ax)}{\sin(bx)}$, where $\{x\}$ denotes the fractional part function is

If f(x) = \begin{cases} A \sin[x] & \text{for } [x] \geq 0 \\ 0 & \text{for } [x] , where $[x]$ denotes the greatest integer less than or equal to $x$, then $\lim_{x \to 0} f(x)$ equals

The value of $\lim_{x \to a} [2 - x + \sqrt{1 + x}]$, where $a \in [0, 1)$ and $[\cdot]$ denotes the greatest integer function is

The value of $\lim_{x \to 0} \frac{x \sin(\sin x) - \sin 2x}{x^6}$ equals

The value of $\lim_{n \to \infty} \frac{a^n + b^n}{a^n - b^n}$, where $a > b > 1$, is

Let f(x) = e^{x^3 + x^2 + x} for any real number x and let g be the inverse function for f. The value of g'(e^3) is

If f(x) = \begin{cases} \frac{\log(1+2ax)-\log(1+bx)}{x} & , x \neq 0 \\ k & , x = 0 \end{cases} is continuous at x=0, then k is equal to

Ex. 12 If $f(x) = [2 + 5|n|\sin x]$, where $n \in \mathbb{I}$ has exactly 9 points of non-derivability in $(0, \pi)$, then possible values of $n$ are (where $[x]$ denotes greatest integer function)

Let f : ℝ → ℝ is a function satisfying f(10 + x) = f(x) and f(2 + x) = f(2 − x), ∀x ∈ ℝ. If f(0) = 101. Then, the minimum possible number of values of x satisfying f(x) = 101, x ∈ [0, 25] is ……….

Let \[f(x) = \frac{e^{\tan x} - e^x}{\tan x - x} + \frac{\log(\sec x + \tan x) - x}{\tan x - x}\] be a continuous function at $x = 0$. The value of $f(0)$ equals

If $ g(x) = \begin{cases} \frac{a^x \log a + a^x}{x \log 2 + x \log a + 1}, & x \neq 0 \\ 2a + x \log 2 + x \log a + 1, & x = 0 \end{cases} $ where $a > 0$, then the value of $a$ for which $g(x)$ is continuous is:

Let $f : \mathbb{R} \to \mathbb{R}$ be such that $f(1) = 3$ and $f'(1) = 6$. Then, $\lim_{x \to 0} \frac{7f(1 + x)}{f(1)}$ equals [2002 AIEEE]

If f(x) = min {1, x², x³}, then

Let h(x) = min{x, x2} for every real number x. Then which of the following is true?

Let $m$ be a positive integer. If $\displaystyle\lim_{x \to 0} |\cos x + \sin 2x + \sin 3x|^{\cot x} = e^m$, then the value of $m$ is:

Let $\lim_{n \to \infty} \frac{1}{2} - \frac{1}{2} \left(1 + \frac{1}{n}\right)^{-1} \times \left(1^1 \times 2^2 \times 3^3 \times \ldots \times n^n\right)^{1/n^2} = e$, where $p$ and $q$ are relatively prime positive integers. Find the value of $|p + q|$.

The value of \[\lim_{x \to \infty} \frac{\int_0^x (\tan^{-1} x)^2 dx}{x^2 + 1}\]

The value of $\lim_{x \to -6} \frac{f(x) - x^2 - 1}{3(x + 6)}$ equals to:

Suppose that $f(0) = 0$ and $f'(0) = 2$. Let $g(x) = f(-f(-f(x)))$. The value of $g'(0)$ is:

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\begin{cases}x^2+3x+a & ,\; x\le 1\\bx+2 & ,\; x>1\end{cases}$ be differentiable on $\mathbb{R}$. Then the value of $\displaystyle\int_{-2}^{2}f(x)\,dx$ equals

Let $g(x)$ be a linear function and $f(x)=\begin{cases}g(x) & ,\; x\le 0\\\left(\dfrac{1+x}{2+x}\right)^{1/x} & ,\; x>0\end{cases}$ is continuous at $x=0$. If $f'(1)=f(-1)$, then the value of $g(3)$ is

If $y^2 = 3\cos^2 x + 2\sin^2 x$, then the value of $y^4 + y\frac{d^2y}{dx^2}$ is

If $f(x) = \begin{cases} \frac{(e^{x(x+3)\ln 27})^{\frac{x}{27}} - 9}{3^x - 27} & ; x 3 \end{cases}$and $\lim_{x \to 3} f(x)$ exists, then $l = $

The value of $\lim_{x \to -\infty} g(x)$ is:

If $f(x) = (x-1)^4(x-2)^3(x-3)^2$, then the value of $f'''(1) + f''(2) + f'(3)$ is:

Let $\lim_{n \to \infty} n \sin\left(\frac{2\pi e}{n}\right) = k\pi$, where $n \in \mathbb{N}$. Find k:

The value of $\displaystyle\lim_{x \to \frac{\pi}{2}} \dfrac{4(x-\pi)\cos^2 x}{\pi(\pi - 2x)\tan\!\left(x - \dfrac{\pi}{2}\right)}$ is equal to:

If $a=\displaystyle\lim_{x\to0}\dfrac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $b=\displaystyle\lim_{x\to0}\dfrac{\sin^2x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $ab^3$ is:

If $\lim_{x \to \infty} \frac{px + q}{qx + p} = 1$ and $\lim_{x \to \infty} \frac{px + q}{qx + p} = m$, where $p, q \neq 0$, then $\lim_{x \to 0}$ is

If $e^y + xy = e$, the ordered pair $\!\left(\dfrac{dy}{dx},\,\dfrac{d^2y}{dx^2}\right)\!$ at $x=0$ is:

If $G(x) = -\sqrt{25 - x^2}$ then $\lim_{x \to 1} \frac{G(x) - G(1)}{x - 1} =$ ______

Limits Questions (1092)