EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Limits>JEE Main - 24 January 2023 Shift 2>Q 76

HARD

JEE Main

IMPORTANT

Earn 100

The set of values of $a$ for which $\lim_{x \to a} ([x - 5] - [2 x + 2]) = 0$ , where, $[ζ]$ denotes the greatest integer less than or equal to $ζ$ is equal to

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Important Questions on Limits

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Limits>JEE Main - 25 January 2023 Shift 1>Q 77

The value of

$\lim_{n \to \infty} \frac{1 + 2 - 3 + 4 + 5 - 6 + \dots + (3 n - 2) + (3 n - 1) - 3 n}{\sqrt{2 n^{4} + 4 n + 3 -} \sqrt{n^{4} + 5 n + 4}}$ is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Limits>JEE Main - 29 January 2023 Shift 1>Q 78

Let

x = 2

be a root of the equation

x^{2} + p x + q = 0

and

f (x) = \{\begin{matrix} \frac{1 - \cos (x^{2} - 4 p x + q^{2} + 8 q + 16)}{{(x - 2 p)}^{4}}, & x \neq 2 p \\ 0, & x = 2 p \end{matrix}

. Then

\lim_{x \to 2 p^{+}} [f (x)]

where

[\cdot]

denotes greatest integer function, is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Limits>JEE Main - 30 January 2023 Shift 1>Q 79

\lim_{x \to 0} \frac{48}{x^{4}} \int_{0}^{x} \frac{t^{3}}{t^{6} + 1} d t

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Limits>JEE Main - 30 January 2023 Shift 2>Q 80

Let

f, g

and

h

be the real valued functions defined on

ℝ

as

f (x) = \{\begin{matrix} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{matrix}, g (x) = \{\begin{matrix} \frac{\sin (x + 1)}{(x + 1)}, & x \neq - 1 \\ 1, & x = - 1 \end{matrix}

and

h (x) = 2 [x] - f (x)

, where

[x]

is the greatest integer

\leq x

. Then the value of

\lim_{x \to 1} g (h (x - 1))

is

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Limits>JEE Main - 31 January 2023 Shift 2>Q 81

\lim_{x \to \infty} \frac{{(\sqrt{3 x + 1} + \sqrt{3 x - 1})}^{6} + {(\sqrt{3 x + 1} - \sqrt{3 x - 1})}^{6}}{{(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}} x^{3}