Limits and Derivatives - Class 11

Let $f(x) = \int_{x}^{x^2} \sqrt{1 - \cos(2\pi t)} \, dt$. If $L_1 = \lim_{x \to 1^+} \frac{f(x)}{(x-1)^2}$ and $L_2 = \lim_{x \to 1^-} \frac{f(x)}{(x-1)^2}$, then the value of $\frac{L_1 - L_2}{\pi \sqrt{2}}$ is

Let $M(x)$ be a $3 \times 3$ matrix whose $(i,j)^{\text{th}}$ entry is given by $m_{ij} = \sin(ijx)$ for $1 \le i, j \le 3$. Let $P_x(\lambda) = \lambda^3 - c_1(x)\lambda^2 + c_2(x)\lambda - c_3(x) = 0$ be the characteristic equation of $M(x)$. Evaluate the limit: $\lim_{x \to 0} \frac{c_3(x)}{x^9}$

Let $f: [0, \infty) \to \mathbb{R}$ be defined by $f(x) = \int_0^x \frac{|\sin t| - \sin t}{2} e^{\cos t} dt$ Let $C = e - \frac{1}{e}$. For each integer $n \ge 1$, let $a_n$ and $b_n$ be the unique positive real numbers satisfying $f(a_n) = 5C - \frac{1}{n} \quad \text{and} \quad f(b_n) = 5C + \frac{1}{n}$ Find the value of $\lim_{n \to \infty} (3a_n - b_n)$.

Evaluate the following limit: $\lim_{n \to \infty} \sum_{k=1}^{2n} \frac{n^3}{k^4} \left( \ln\left(1 + \frac{k^2}{n^2}\right) - \frac{k^2}{n^2} \right)$

Let $f(x) = \sqrt{x^2 - 1 - 2ix} + \sqrt{x^2 - 1 + 2ix}$, where $\sqrt{z}$ denotes the principal square root of $z$ (i.e., $-\pi < \text{Arg}(z) \le \pi$). Evaluate the limit: $L = \lim_{x \to 0^-} \left( \cos x + f(x) \right)^{\frac{1}{\sin x}}$

Let $L = \lim_{x \to 0^+} \frac{1}{x^3} \int_0^x \frac{\sin^2 t \cdot \sin^2(x-t)}{(\tan t + \tan(x-t))^2} e^{t x} \, dt$ Find the value of $L$.

Let a curve $C$ be defined parametrically by $x(t) = (t-1)^3$ $y(t) = \lim_{n \to \infty} \frac{t^{2n} (t-1)^3 (t^2 + 4) + |t-1|^3 (t^3 + 1)}{t^{2n} + 1}$ for $t \in (0, \infty)$. Let $L = \lim_{x \to 0^+} \frac{dy}{dx}$ and $R = \lim_{x \to 0^-} \frac{dy}{dx}$. Find the value of $10L + R$.

Evaluate the following limit: $\lim_{n \to \infty} n^2 \left( \frac{2^n - \sum_{k=0}^{n-4} \binom{n}{k}}{n^3} - \frac{1}{6} \right)$

Let $y = f(x)$ be the solution to the initial value problem $(x^2 - 1) \frac{dy}{dx} + 2x y = 3x^2 + 1, \quad f(2) = 4$ defined on its maximal interval of existence. Find the value of $\lim_{x \to -1} f(x)$.

Let $f(x) = \frac{1}{x^3} \begin{vmatrix} \sin x & \sin 2x & \sin 3x \\ x+[x] & 3x+[2x] & 5x+[3x] \\ 1 & 3 & 5 \end{vmatrix}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Evaluate $\lim_{x \to 0} f(x)$.

Let $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$. Using differentials, the approximate value of $f(0.98) + f(1.02)$ is:

Let $y = f(x)$ be the function representing the rectangular hyperbola $xy = 8$ in the first quadrant. Let $P(x_0, y_0)$ be a point on this hyperbola such that $\lim_{x \to x_0} \frac{x^{10} (f(x))^8 - x_0^{10} (f(x_0))^8}{x^5 (f(x))^9 - x_0^5 (f(x_0))^9} = -256$ Let $N$ be the normal to the hyperbola at the point $P$. If $N$ intersects the coordinate axes at points $A$ and $B$, find the area of the triangle $OAB$, where $O$ is the origin.

Let $S_k(n) = \sum_{i=1}^n i^k$. Evaluate the limit: $\lim_{N \to \infty} \sum_{n=1}^N \frac{2 S_1(n) + 1}{4 S_3(n)}$

Let $y = f(x)$ be a differentiable function on $(-1, 1)$ satisfying the differential equation $\left( \cosh(y) + \frac{1}{\sqrt{1-y^2}} + \left( \ln(y + \sqrt{1+y^2}) \right)^2 \right) \frac{dy}{dx} = \frac{x^2}{\sqrt{1+x^2}} + \cos(x) + e^{x^2}$ with the initial condition $f(0) = 0$. Evaluate the limit: $\lim_{x \to 0} \frac{1}{x^5} \int_{-x}^{x} \frac{t^2 f(t) + t^3 \sin(3t)}{e^{(f(t))^2}} dt$

Let $L = \lim_{x \to 0} \left( \sum_{k=0}^{\lfloor 1/x^2 \rfloor} (24 x \cdot \text{sgn}(x))^k \right)^{\frac{4}{x} \sum_{n=1}^{\infty} \frac{1}{n(n+2)}}$ Find the value of $L$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function and $\text{sgn}(\cdot)$ is the signum function.

Let $A$ be a $3 \times 3$ real skew-symmetric matrix such that $\text{Tr}(A A^T) = 28$. Let $M(x) = I + \sin(x) A + \tan^2(x) A^T A$, where $I$ is the $3 \times 3$ identity matrix. Evaluate the limit: $L = \lim_{x \to 0} \left( \det(M(x)) \right)^{\frac{1}{x^2}}$

Evaluate the following limit: $\lim_{x \to 0} \left( \frac{1}{x^2} \int_0^{x^2} \frac{1+2t^2}{\sqrt{1+t^2}} \, dt \right)^{\csc^4 x}$

Let $S_n = \sum_{k=0}^{n} \binom{n+k-1}{k} \frac{1}{n^k}$. Evaluate the limit: $\lim_{n \to \infty} \left( \frac{S_n}{e} \right)^n$

Let $f(x) = \lim_{n \to \infty} \frac{1}{n^4} \sum_{r=1}^{n} \left( \left[ r^3 x \right] + r \left[ r x \right]^2 + r^2 \left[ r x \right] \right)$, where $[ \cdot ]$ denotes the greatest integer function. If $x > 0$ and $f(x) = 6$, find the value of $x$.

Let $B_n = \begin{pmatrix} \sum_{r=1}^n \frac{1}{\sqrt{n^2+r}} & \sum_{r=1}^n \frac{r^2}{n^3+r^2} \\ \sum_{r=1}^n \frac{r}{n^2+r} & \sum_{r=1}^n \frac{1}{\sqrt{4n^2+r}} \end{pmatrix}$ and $A = \lim_{n \to \infty} (3I - B_n)^{-1}$, where $I$ is the $2 \times 2$ identity matrix. Find the value of $\lim_{k \to \infty} \det(A^k)$.

Let $\vec{u}(x)$ and $\vec{v}(x)$ be two vector-valued functions defined for $x \neq 0$ as: $\vec{u}(x) = |x| \sin\left(\frac{1}{x}\right) \hat{i} + |x| \cos\left(\frac{1}{x}\right) \hat{j} - \left( |x| \left(\sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right)\right) + x^2 \sin\left(\frac{1}{x^2}\right) \right) \hat{k}$ $\vec{v}(x) = |x| \cos\left(\frac{1}{x}\right) \hat{i} - |x| \sin\left(\frac{1}{x}\right) \hat{j} + |x| \left(\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right)\right) \hat{k}$ If $A(x)$ represents the area of the triangle formed by the vectors $\vec{u}(x)$ and $\vec{v}(x)$, evaluate: $\lim_{x \to 0} \frac{A(x)}{x^2}$

Evaluate the following limit: $\lim_{x \to \infty} x^4 \left( \arctan(x+1) - \arctan(x-1) - \frac{2}{1 + \left(x + e^{-x} \text{sgn}(\sin x)\right)^2} \right)$

Let $I(\lambda) = \int_0^1 \frac{\lambda}{\lambda^2 + x^2} \ln\left(\frac{1}{x}\right) e^{-x} \, dx$. Evaluate the limit $\lim_{\lambda \to 0^+} \frac{I(\lambda)}{-\ln \lambda}$

Let $f(x) = (\cos x + i \sin x) \left( \frac{\sqrt{1+\sin 2x} + i\sqrt{1-\sin 2x}}{\sqrt{1+\sin 2x} - i\sqrt{1-\sin 2x}} \right)^{\sin x}$ for $x \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right)$, where $i = \sqrt{-1}$. Evaluate the limit: $\lim_{h \to 0} \frac{\left| f\left(\frac{\pi}{2} + h\right) - f\left(\frac{\pi}{2}\right) \right|}{|h|}$

Let $y: [1, e] \to \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$ be a differentiable function satisfying the differential equation $\lim_{h \to 0} \frac{\arcsin(\sin(y(x+h))) - \arcsin(\sin(y(x)))}{h} + \frac{y(x)}{x} = \frac{y^2(x)}{x^2}$ for all $x \in [1, e]$, with the initial condition $y(1) = \pi$. Find the value of $y(e)$.

Let $\vec{u}(t)$ and $\vec{v}(t)$ be two differentiable vector-valued functions in $\mathbb{R}^3$. At $t=0$, the following properties are given: $|\vec{u}(0)| = 2$, $|\vec{v}(0)| = 3$, and the angle between $\vec{u}(0)$ and $\vec{v}(0)$ is $\cos^{-1}\left(\frac{2}{3}\right)$. The rates of change of their magnitudes at $t=0$ are $3$ and $-1$ respectively. Additionally, the rate of change of their scalar product at $t=0$ is $5$. Evaluate the limit: $\lim_{h \to 0} \frac{|\vec{u}(h) \times \vec{v}(h)|^2 - |\vec{u}(0) \times \vec{v}(0)|^2}{h}$

Let $f: (0, 7) \to \mathbb{R}$ be a function defined as $f(x) = |x^2 - 7x + 6| ([x] - \{x\})$, where $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}$ denotes the fractional part of $x$. Let $S$ be the set of all points in $(0, 7)$ where $f(x)$ is not differentiable. Find the value of $\sum_{x \in S} x$.

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $B = \begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}$, and $C = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$. Let $F(x) = A + xB + x^2C$ be a matrix-valued function of a real variable $x$. If $H(x) = (F(x))^2$, find the value of the matrix $H''(0)$.

Let $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \tan^{-1}\left(\frac{1+x}{1-x}\right)$. Evaluate the limit $\lim_{x \to \infty} \int_0^x f'(t) \, dt$.

Let $O_1$ and $D_1$ be the order and degree of the differential equation representing the family of curves $y = c e^x + c^3$, where $c$ is an arbitrary real constant. Let $O_2$ and $D_2$ be the order and degree of the differential equation of the orthogonal trajectories of this family. Find the value of $O_1 + D_1 + O_2 + D_2$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = 12|x^2 - 4| + 4|x|^3 - 30x^2 + 72|x|$. If $S$ is the set of all points of local extrema of $f$, then the value of $\sum_{x \in S} 2^{|x|}$ is equal to

Let $S$ be the set of all real values of $x$ for which the function $f(x) = \left| x^2 - 4 \right| e^{-|x|}$ is strictly increasing. Which of the following represents the set $S$?

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x,y) = \begin{cases} \frac{x^3 y - x y^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$ Let $H(x,y) = \sqrt{|f(x,y)|}$. Let $D_{\vec{u}} H(0,0) = \lim_{t \to 0^+} \frac{H(t\cos\theta, t\sin\theta) - H(0,0)}{t}$ denote the directional derivative of $H$ at $(0,0)$ in the direction of the unit vector $\vec{u} = (\cos\theta, \sin\theta)$ for $\theta \in [0, 2\pi)$. Let $M$ be the maximum value of $D_{\vec{u}} H(0,0)$ over all such unit vectors, and let $L$ be the number of distinct unit vectors for which this maximum is attained. Find the value of $L \times M$.

Let $f(x) = \arcsin\left( \frac{3\sin(2x) + 4\cos(2x)}{5} \right) + \arccos\left( \frac{5\cos(3x) - 12\sin(3x)}{13} \right)$. Find the value of $f'\left( \frac{\pi}{2} \right)$.

Let $f(x) = \log_{([x] + \{x\}^3)} (x^3)$ for $x > 1$, where $[x]$ and $\{x\}$ denote the greatest integer and fractional part functions respectively. If $L$ and $R$ are the left-hand and right-hand derivatives of $f(x)$ at $x=2$, then the value of $R - L$ is:

Let $f: D \to \mathbb{R}$ be a real-valued function defined by $f(x) = \ln\left( \sqrt{x^2 - 25} - x \right) - \frac{x}{6}$, where $D$ is the maximal possible domain of $f$. If $m$ is the slope of the normal to the curve $y = f(x)$ at $x = 13$, then the value of $m$ is:

Let $y(x)$ be the real-valued continuous function defined implicitly by the equation $y^5 + y^3 + y = |x^2 - 4|^{|x-1|}$ for all $x \in \mathbb{R}$. Let $S$ be the set of all points $x \in \mathbb{R}$ where $y(x)$ is not differentiable. Find the sum of all elements in $S$.

Let $a_0, a_1, a_2, \dots, a_{402}$ be real numbers such that $\sum_{r=0}^{402} a_r x^r = (1 - 4x + x^2)^{201}$ for all $x \in \mathbb{R}$. Let $f: D \to \mathbb{R}$ be defined by $f(x) = \ln \left( \sum_{r=0}^{402} a_r x^r \right)$ where $D \subseteq \mathbb{R}$ is the maximal domain of $f$. Find the value of $f'(1)$.

Let $f(x) = \begin{vmatrix} 0 & x & 1 & 241 \\ -x & 0 & 1 & x \\ -1 & -1 & 0 & x^2 \\ -241 & -x & -x^2 & 0 \end{vmatrix}$.

If $g(x) = (f(x))^{1 + \sin^2(\pi x)}$, find the value of $g'(1)$.

Let $f: (0, \infty) \to \mathbb{R}$ be defined as $f(x) = \int_0^{x^2} |t - x| e^t \, dt$ Let $L = \lim_{x \to 1^-} f''(x)$ and $R = \lim_{x \to 1^+} f''(x)$. Find the value of $R - L$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \int_0^x (t^2 - 6t + 8)(|t-3| - 2) \,dt$ Which of the following represents the complete set of intervals where $f(x)$ is strictly increasing?

Let $A(x) = \begin{bmatrix} x^3+1 & x^2+x & x+1 \\ 9 & 6 & 3 \\ 28 & 12 & 4 \end{bmatrix}$. Let $f(x) = \sqrt{\det(A(x) A^T(x))}$. If $L$ and $R$ denote the left-hand derivative and right-hand derivative of $f(x)$ at $x=2$ respectively, find the value of $R - L$.

A particle moves along a straight line such that its position $x(t)$ at time $t$ is given by $x(t) = \ln(t^2 - 8t + 20) - |t-4|$ Let $N$ be the number of points in the time interval $t \in (1, 7)$ where the particle comes to instantaneous rest (i.e., $\frac{dx}{dt} = 0$). Let $M$ be the number of points in the same interval where $x(t)$ attains a local extremum. The value of $N + M$ is:

The velocity $v(t)$ of a particle moving along the x-axis is given by $v(t) = \frac{\ln(1+t) \cdot (e^t - 1) - t^2}{t^4}$ for $t \neq 0$. If the acceleration $a(t)$ of the particle is continuous at $t=0$, what is the value of $a(0)$?

A composite reflective surface is formed by the parabola $y^2 = 4(x+4)$ for $x < 0$ and the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ for $x \ge 0$. A light ray is emitted from the point $S(-3, 0)$ and strikes the parabolic surface at a point $P$ in the first quadrant (where $y > 0$). The ray reflects off the parabola, travels parallel to the x-axis, and then strikes the elliptical surface at a point $Q$. The ray reflects off the ellipse and intersects the x-axis at a point $R$. Find the x-coordinate of $R$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^3 - 3x^2 + 3x$. Let $g(x)$ be the inverse function of $f(x)$. Consider the integral $I = \int_{0}^{2} \frac{(g''(x))^2}{(g'(x))^3} \, dx$ Which of the following is the correct value of $I$?

Let $\omega = e^{i 2\pi/5}$ and define the polynomial $P(x) = \prod_{k=1}^4 (x - \omega^k)$ for $x \in \mathbb{R}$. Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \sqrt{\max(0, 1 - P(x))}$ and $g: \mathbb{R} \to \mathbb{R}$ be defined as $g(x) = x^2$. If $m$ and $n$ are the number of points of non-differentiability of the composite functions $f(g(x))$ and $g(f(x))$ respectively, find the value of $m + n$.

Let $M(x) = \begin{pmatrix} x & |x|^3 \\ 0 & -x \end{pmatrix}$. Let $Y(x)$ be a $2 \times 2$ matrix-valued function satisfying the differential equation $\frac{dY}{dx} = M(x) Y(x)$ with the initial condition $Y(0) = I$, where $I$ is the $2 \times 2$ identity matrix. Let $f(x)$ be the entry in the first row and second column of $Y(x)$. If $k$ is the maximum integer such that $f(x)$ is $k$-times continuously differentiable on $\mathbb{R}$, find the ordered pair $\left(k, f(\sqrt{2})\right)$.

Evaluate the following limit: $\lim_{x \to 0^-} \frac{\ln(1 + 3x - 2\sqrt{x^2 + x^4}) - 3x + 2\sqrt{x^2}}{x^2}$

Let $f: \left(0, \frac{\pi}{4}\right) \to (0, \infty)$ be a continuous function such that for all $x \in \left(0, \frac{\pi}{4}\right)$, $(\sin x)^{3x} \le f(x) \le (\tan x)^{2x}$ Let $S$ be the set of all possible values of the limit $L = \lim_{x \to 0^+} (f(x))^{\frac{1}{x \ln x}}$ assuming the limit exists. Find the supremum of the set $S$.