Limits XII — JEE Advanced PYQs

21 previous year questions on Limits XII (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let $α$ and $β$ be the real numbers such that
$x \to 0 lim \frac{1}{x ^{3}} (\frac{α}{2} \int_{0}^{x} \frac{1}{1 - t ^{2}} d t + β x cos x) = 2$

Then the value of $α + β$ is ____.
JEE Advanced 2025View solution →
Let $k \in R$ . If $lim_{x \to 0^{+}} (sin (sin k x) + cos x + x)^{\frac{2}{x}} = e^{6}$ , then the value of $k$ is
JEE Advanced 2024View solution →
Let $f : (0, 1) \to R$ be the functions defined as $f (x) = n$ if $x \in [\frac{1}{n + 1}, \frac{1}{n})$ where $n \in N$ . Let $g : (0, 1) \to R$ be a …
JEE Advanced 2023View solution →
Let $α$ be a positive real number. Let $f : R \to R$ and $g : (α, \infty) \to R$ be the functions defined by …
JEE Advanced 2022View solution →
If
$β = x \to 0 lim \frac{e ^{x^{3}} - ( 1 - x ^{3} ) ^{\frac{1}{3}} + (( 1 - x ^{2} ) ^{\frac{1}{2}} - 1 ) sin x}{x sin ^{2} x}$

then the value of $6 β$ is ____.
JEE Advanced 2022View solution →
Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit
$x \to 0^{+} lim \frac{( 1 - x ) ^{\frac{1}{x}} - e ^{- 1}}{x ^{a}}$

is equal to a nonzero real number, is ____.
JEE Advanced 2020View solution →
The value of the limit
$x \to \frac{π}{2} lim \frac{4 2 ( sin 3 x + sin x )}{( 2 sin 2 x sin \frac{3 x}{2} + cos \frac{5 x}{2} ) - ( 2 + 2 cos 2 x + cos \frac{3 x}{2} )}$

is ____
JEE Advanced 2020View solution →
For $a \in R, ∣ a ∣ > 1$ , let …
JEE Advanced 2019View solution →
For each positive integer $n$ , let $y_{n} = \frac{1}{n} ((n + 1) (n + 2) \dots (n + n))^{\frac{1}{n}}$ . For $x \in R$ , let $[x]$ be the greatest integer less than or equal to $x$ . If $lim_{n \to \infty} y_{n} = L$ , then the value of $[L]$ is ____.
JEE Advanced 2018View solution →
Let $α, β \in R$ be such that $lim_{x \to 0} \frac{x ^{2} s i n ( β x )}{α x - s i n x} = 1$ . Then $6 (α + β)$ equals
JEE Advanced 2016View solution →
Let $f (x) = lim_{n \to \infty} (\frac{n ^{n} ( x + n ) ( x + \frac{n}{2} ) \dots ( x + \frac{n}{n} )}{n ! ( x ^{2} + n ^{2} ) ( x ^{2} + \frac{n ^{2}}{4} ) \dots ( x ^{2} + \frac{n ^{2}}{n ^{2}} )})^{\frac{x}{n}}$ , for all $x > 0$ . Then
JEE Advanced 2016View solution →
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 $\frac{m}{n}$ is
JEE Advanced 2015View solution →
The largest value of the nonnegative integer $a$ for which
$x \to 1 lim {\frac{- a x + sin ( x - 1 ) + a}{x + sin ( x - 1 ) - 1}}^{\frac{1 - x}{1 - x}} = \frac{1}{4}$

is
JEE Advanced 2014View solution →
For $a \in R$ (the set of all real numbers), $a \neq = - 1$ ,
$lim_{n \to \infty} \frac{( 1 ^{a} + 2 ^{a} + \dots + n ^{a} )}{( n + 1 ) ^{a - 1} [( na + 1 ) + ( na + 2 ) + \dots + ( na + n )]} = \frac{1}{60}$
Then $a =$
JEE Advanced 2013View solution →
Let $α (a)$ and $β (a)$ be the roots of the equation $(3 1 + a - 1) x^{2} + (1 + a - 1) x + (6 1 + a - 1) = 0$ where $a > - 1$ .
Then $lim_{a \to 0^{+}} α (a)$ and $lim_{a \to 0^{+}} β (a)$ are
JEE Advanced 2012View solution →
If
$x \to 0 lim [1 + x ln (1 + b^{2})]^{\frac{1}{x}} = 2 b sin^{2} θ, b > 0 and θ \in (- π, π]$

then the value of $θ$ is
JEE Advanced 2011View solution →
The value of $lim_{x \to 0} \frac{1}{x ^{3}} \int_{0}^{x} \frac{t l n ( 1 + t )}{t ^{4} + 4} d t$ is
JEE Advanced 2010View solution →
Let
$L = x \to 0 lim \frac{a - a ^{2} - x ^{2} - \frac{x ^{2}}{4}}{x ^{4}}, a > 0.$

If $L$ is finite, then
JEE Advanced 2009View solution →
Let
$g (x) = \frac{( x - 1 ) ^{n}}{l o g c o s ^{m} ( x - 1 )}; 0 < x < 2, m$ and $n$ are integers, $m \neq = 0, n > 0$ , and let $p$ be the left hand derivative of $∣ x - 1∣$ at $x = 1$ .
If $lim_{x \to 1 +} g (x) = p$ , then
JEE Advanced 2008View solution →
$lim_{x \to \frac{π}{4}} \frac{\int _{2}^{s e c^{2} x} f ( t ) d t}{x ^{2} - \frac{π ^{2}}{16}}$ equals
JEE Advanced 2007View solution →
For $x > 0, lim_{x \to 0} ((sin x)^{1/ x} + (1/ x)^{s i n x})$ is
JEE Advanced 2006View solution →