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Limits Continuity and Differentiability

52 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Limits Continuity and Differentiability

Limits Continuity and Differentiability

52 questions · Mathematics · JEE Advanced

Let $R$ denote the set of all real numbers. Define the function $f : R \to R$ by $f (x) = {2 - 2 x^{2} - x^{2} sin \frac{1}{x} 2 if x e q 0, if x = 0.$ …2025 · Shift 1 · Q19 · MCQ
Let α and β be the real numbers such that $x \to 0 lim \frac{1}{x ^{3}} (\frac{α}{2} 0 \int x \frac{1}{1 - t ^{2}} d t + β x cos x) = 2.$ Then the value of α + β is .2025 · Shift 1 · Q27 · Numerical
Let $R$ denote the set of all real numbers. For a real number $x$ , let [ x ] denote the greatest integer less than or equal to $x$ . Let $n$ denote a natural number. Match each entry in List-I to the correct entry in List-II and… Includes table2025 · Shift 1 · Q31 · MCQ
Let $x_{0}$ be the real number such that $e^{x_{0}} + x_{0} = 0$ . For a given real number $α$ , define $g (x) = \frac{3 x e ^{x} + 3 x - α e ^{x} - α x}{3 ( e ^{x} + 1 )}$ for all real numbers $x$ . Then which one of the following statements is…2025 · Shift 2 · Q17 · MCQ
Let $f : R \to R$ and $g : R \to R$ be functions defined by $f (x) = {x ∣ x ∣ sin (\frac{1}{x}), 0, x e q 0, x = 0, and g (x) = {1 - 2 x, 0, 0 \leq x \leq \frac{1}{2}, otherwise .$ … Includes table2024 · Shift 1 · Q34 · MCQ
Let $k \in R$ . If $x \to 0 + lim (sin (sin k x) + cos x + x)^{\frac{2}{x}} = e^{6}$ , then the value of $k$ is2024 · Shift 2 · Q20 · MCQ
Let $f : R \to R$ be a function defined by $f (x) = {x^{2} sin (\frac{π}{x ^{2}}), 0, if x e q 0, if x = 0.$ Then which of the…2024 · Shift 2 · Q21 · MCQ
Let $S$ be the set of all $(α, β) \in R \times R$ such that $x \to \infty lim \frac{s i n ( x ^{2} ) ( l o g _{e} x ) ^{α} s i n ( \frac{1}{x ^{2}} )}{x ^{α β} ( l o g _{e} ( 1 + x ) ) ^{β}} = 0.$ …2024 · Shift 2 · Q22 · Multiple correct
Let $f : (0, 1) \to R$ be the function defined as $f (x) = [4 x] (x - \frac{1}{4})^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following statements…2023 · Shift 2 · Q23 · Multiple correct
Let $α$ be a positive real number. Let $f : R \to R$ and $g : (α, \infty) \to R$ be the functions defined by $f (x) = sin (\frac{π x}{12}) and g (x) = \frac{2 l o g _{e} ( x - α )}{l o g _{e} ( e ^{x} - e ^{α} )} .$ …2022 · Shift 1 · Q20 · Numerical
If $β = x \to 0 lim \frac{e ^{x^{3}} - ( 1 - x ^{3} ) ^{\frac{1}{3}} + ( ( 1 - x ^{2} ) ^{\frac{1}{2}} - 1 ) s i n x}{x s i n ^{2} x},$ then the value of $6 β$ is .2022 · Shift 2 · Q23 · Numerical
For positive integer $n$ , define $f (n) = n + \frac{16 + 5 n - 3 n ^{2}}{4 n + 3 n ^{2}} + \frac{32 + n - 3 n ^{2}}{8 n + 3 n ^{2}} + \frac{48 - 3 n - 3 n ^{2}}{12 n + 3 n ^{2}} + \dots + \frac{25 n - 7 n ^{2}}{7 n ^{2}} .$ Then, the value of $n \to \infty lim f (n)$ …2022 · Shift 2 · Q36 · MCQ
Let f : R $\to$ R be defined by $f (x) = \frac{x ^{2} - 3 x - 6}{x ^{2} + 2 x + 4}$ Then which of the following statements is (are) TRUE?2021 · Shift 1 · Q31 · Multiple correct
Let the function f : R $\to$ R be defined by f(x) = x3 $-$ x2 + (x $-$ 1)sin x and let g : R $\to$ R be an arbitrary function. Let fg : R $\to$ R be the product function defined by (fg)(x) = f(x)g(x). Then which of the following statements…2020 · Shift 1 · Q25 · Multiple correct
let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit $x \to 0^{+} lim \frac{( 1 - x ) ^{1/ x} - e ^{- 1}}{x ^{a}}$ is equal to a non-zero real number,…2020 · Shift 1 · Q36 · Numerical
Let the functions $f : (- 1, 1) \to R$ and $g : (- 1, 1) \to (- 1, 1)$ be defined by $f (x) = ∣2 x - 1∣ + ∣2 x + 1∣$ and $g (x) = x - [x]$ , where [x] denotes the greatest integer less than or equal to x. Let $f o g : (- 1, 1) \to R$ be the…2020 · Shift 2 · Q23 · Numerical
The value of the limit $x \to \frac{π}{2} lim \frac{4 2 ( s i n 3 x + s i n x )}{( 2 s i n 2 x s i n \frac{3 x}{2} + c o s \frac{5 x}{2} ) - ( 2 + 2 c o s 2 x + c o s \frac{3 x}{2} )}$ …2020 · Shift 2 · Q24 · Numerical
Let f : R $\to$ R and g : R $\to$ R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y $\in$ R. If $x \to 0 lim g (x) = 1$ , then which of the following statements is/are TRUE?2020 · Shift 2 · Q27 · Multiple correct
Let f : R $\to$ R be given by $f (x) = ⎩ ⎨ ⎧ x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 3 x + 1, x^{2} - x + 1, \frac{2}{3} x^{3} - 4 x^{2} + 7 x - \frac{8}{3}, (x - 2) lo g_{e} (x - 2) - x + \frac{10}{3}, x < 0; 0 \leq x < 1; 1 \leq x < 3; x \geq 3; ⎭ ⎬ ⎫$ …2019 · Shift 1 · Q30 · Multiple correct
For $a \in R, ∣ a ∣ > 1$ , let $n \to \infty lim (\frac{1 + 3 2 + ... 3 n}{n ^{7/3} ( \frac{1}{( an + 1 ) ^{2}} + \frac{1}{( an + 2 ) ^{2}} + ... + \frac{1}{( an + n ) ^{2}} )}) = 54$ …2019 · Shift 2 · Q23 · Multiple correct
Let f : R be a function. We say that f has PROPERTY 1 if $h \to 0 lim \frac{f ( h ) - f ( 0 )}{∣ h ∣}$ exists and is finite, and PROPERTY 2 if $h \to 0 lim \frac{f ( h ) - f ( 0 )}{h ^{2}}$ …2019 · Shift 2 · Q24 · Multiple correct
For every twice differentiable function $f : R \to [- 2, 2]$ with $(f (0))^{2} + (f^{'} (0))^{2} = 85$ , which of the following statement(s) is(are) TRUE?2018 · Shift 1 · Q22 · Multiple correct
Let f : R $\to$ R and g : R $\to$ R be two non-constant differentiable functions. If f'(x) = (e(f(x) $-$ g(x))) g'(x) for all x $\in$ R and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?2018 · Shift 1 · Q23 · Multiple correct
The value of $((lo g_{2} 9)^{2})^{\frac{1}{l o g _{2} ( l o g _{2} 9 )}} \times (7)^{\frac{1}{l o g _{4} 7}}$ is ....................2018 · Shift 1 · Q25 · Numerical
Let f : (0, $π$ ) $\to$ R be a twice differentiable function such that $t \to x lim \frac{f ( x ) s i n t - f ( t ) s i n x}{t - x} = sin^{2} x$ for all x $\in$ (0, $π$ ). If $f (\frac{π}{6}) = - \frac{π}{12}$ …2018 · Shift 2 · Q24 · Multiple correct
Let $f_{1} : R \to R$ , $f_{2} : (- \frac{π}{2}, \frac{π}{2}) \to R$ , $f_{3} : (- 1, e^{π /2} - 2) \to R$ and $f_{4} : R \to R$ be functions defined by (i) $f_{1} (x) = sin (1 - e^{- x^{2}})$ …2018 · Shift 2 · Q36 · MCQ
Let f : R $\to$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?2017 · Shift 1 · Q20 · Multiple correct
Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $f (x) = x cos (π (x + [x]))$ is discontinuous?2017 · Shift 1 · Q23 · Multiple correct
Let f : R $\to$ R be a differentiable function such that f(0) = 0, $f (\frac{π}{2}) = 3$ and f'(0) = 1. If $g (x) = x \int π /2 [f^{'} (t) cosec t - cot t cosec t f (t)] d t$ for $x \in (0, \frac{π}{2}]$ …2017 · Shift 1 · Q28 · Numerical
If f : R $\to$ R is a twice differentiable function such that f"(x) > 0 for all x $\in$ R, and $f (\frac{1}{2}) = \frac{1}{2}$ , f(1) = 1, then2017 · Shift 2 · Q19 · MCQ
Let $f (x) = \frac{1 - x ( 1 + ∣1 - x ∣ )}{∣1 - x ∣} cos (\frac{1}{1 - x})$ for x $e$ 1. Then2017 · Shift 2 · Q30 · Multiple correct
Let $α$ , $β \in$ R be such that $x \to 0 lim \frac{x ^{2} s i n ( β x )}{α x - s i n x} = 1$ . Then 6( $α$ + $β$ ) equals .2016 · Shift 1 · Q36 · Numerical
Let a, b $\in$ R and f : R $\to$ R be defined by $f (x) = a cos (∣ x^{3} - x ∣) + b ∣ x ∣ sin (∣ x^{3} + x ∣)$ . Then f is2016 · Shift 2 · Q34 · Multiple correct
Let $f : [- \frac{1}{2}, 2] \to R$ and $g : [- \frac{1}{2}, 2] \to R$ be function defined by $f (x) = [x^{2} - 3]$ and $g (x) = ∣ x ∣ f (x) + ∣4 x - 7∣ f (x)$ , where [y] denotes the greatest integer less than or equal…2016 · Shift 2 · Q36 · Multiple correct
Let $g : R \to R$ be a differentiable function with $g (0) = 0$ , $g^{'} (0) = 0$ and $g^{'} (1) e 0$ . Let $f (x) = {\frac{x}{∣ x ∣} g (x), 0, x e 0 x = 0$ and $h (x) = e^{∣ x ∣}$ …2015 · Shift 1 · Q39 · Multiple correct
Let m and n be two positive integers greater than 1. If $α \to 0 lim (\frac{e ^{c o s (α^{n})} - e}{α ^{m}}) = - (\frac{e}{2})$ then the value of $\frac{m}{n}$ …2015 · Shift 2 · Q40 · Numerical
Let $f : (a, b) \to [1, \infty)$ be a continuous function and g : R $\to$ R be defined as $g (x) = {0, x b$ Then,2014 · Shift 1 · Q32 · Multiple correct
The largest value of the non-negative integer a for which $x \to 1 lim {\frac{- a x + s i n ( x - 1 ) + a}{x + s i n ( x - 1 ) - 1}}^{\frac{1 - x}{1 - x}} = \frac{1}{4}$ is2014 · Shift 1 · Q39 · Numerical
Let f : R $\to$ R and g : R $\to$ R be respectively given by f(x) = | x | + 1 and g(x) = x2 + 1. Define h : R $\to$ R by $h (x) = {max {f (x), g (x)}, min {f (x), g (x)}, i f x \leq 0. i f x > 0.$ …2014 · Shift 1 · Q40 · Numerical
$a \in R$ (the set of all real numbers), a $e -$ 1, $n \to \infty lim \frac{( 1 ^{a} + 2 ^{a} + ... + n ^{a} )}{( n + 1 ) ^{a - 1} [( na + 1 ) + ( na + 2 ) + ... + ( na + n )]} = \frac{1}{60}$ , Then a = ?2013 · Shift 2 · Q38 · Multiple correct
If $x \to \infty lim (\frac{x ^{2} + x + 1}{x + 1} - a x - b) = 4$ , then2012 · Shift 1 · Q36 · MCQ
Let $f (x) = {x^{2} cos \frac{π}{x}, 0, x e 0 x = 0$ x $\in$ R, then f is2012 · Shift 1 · Q38 · MCQ
For every integer n, let an and bn be real numbers. Let function f : R $\to$ R be given by $f (x) = {a_{n} + sin π x, b_{n} + cos π x, f or x \in [2 n, 2 n + 1] f or x \in (2 n - 1, 2 n)$ …2012 · Shift 2 · Q38 · Multiple correct
Let f : R $\to$ R be a function such that $f (x + y) = f (x) + f (y), \forall x, y \in R$ . If f(x) is differentiable at x = 0, then2011 · Shift 1 · Q41 · Multiple correct
If $x \to 0 lim [1 + x ln (1 + b^{2})]^{1/ x} = 2 b sin^{2} θ$ , $b > 0$ and $θ \in (- π, π]$ , then the value of $θ$ is2011 · Shift 2 · Q33 · MCQ
If $f (x) = {- x - \frac{π}{2}, - cos x x \leq - \frac{π}{2} - \frac{π}{2} 1$ , then2011 · Shift 2 · Q36 · Multiple correct
Let $L = x \to 0 lim \frac{a - a ^{2} - x ^{2} - \frac{x ^{2}}{4}}{x ^{4}}, a > 0$ . If L is finite, then2009 · Shift 1 · Q37 · Multiple correct
Let $f (x)$ be a non-constant twice differentiable function defined on $(- \infty, \infty)$ such that $f (x) = f (1 - x)$ and $f^{'} (\frac{1}{4}) = 0.$ Then,2008 · Shift 1 · Q32 · Multiple correct
Let $g (x) = \frac{( x - 1 ) ^{n}}{l o g c o s ^{m} ( x - 1 )}; 00$ , and let $p$ be the left hand derivative of $∣ x - 1∣$ at $x = 1$ . If $x \to 1^{+} lim g (x) = p$ , then2008 · Shift 1 · Q44 · MCQ
Consider the function $f : (- \infty, \infty) \to (- \infty, \infty)$ defined by $f (x) = \frac{x ^{2} - a x + 1}{x ^{2} + a x + 1}, 0 < a < 2.$ Which of the following is true?2008 · Shift 2 · Q23 · MCQ
Let the function $g : (- \infty, \infty) \to (- \frac{π}{2}, \frac{π}{2})$ be given by $g (u) = 2 tan^{- 1} (e^{u}) - \frac{π}{2} .$ Then, $g$ is2008 · Shift 2 · Q34 · MCQ
In the following [x] denotes the greatest integer less than or equal to x. Match the functions in Column I with the properties Column II. Includes table2007 · Shift 1 · Q43 · MCQ