(A) We need to find $\lim_{x \rightarrow 1} g(h(x-1))$. Let $t = x-1$. As $x \rightarrow 1$,$t \rightarrow 0$. So,we evaluate $\lim_{t \rightarrow 0}

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(A) We need to find $\lim_{x \rightarrow 1} g(h(x-1))$. Let $t = x-1$. As $x \rightarrow 1$,$t \rightarrow 0$. So,we evaluate $\lim_{t \rightarrow 0} g(h(t))$.$h(t) = 2[t] - f(t)$.For $t \rightarrow 0^-$,$[t] = -1$ and $f(t) = \frac{t}{|t|} = -1$. Thus,$h(t) = 2(-1) - (-1) = -2 + 1 = -1$.Then,$\lim_{t \rightarrow 0^-} g(h(t)) = g(-1) = 1$.For $t \rightarrow 0^+$,$[t] = 0$ and $f(t) = \frac{t}{|t|} = 1$. Thus,$h(t) = 2(0) - 1 = -1$.Then,$\lim_{t \rightarrow 0^+} g(h(t)) = g(-1) = 1$.Since the left-hand limit and right-hand limit are equal,the limit is $1$.

Class 12 Mathematics Relation and Function Mix Examples of Relation and Function

Difficult

Let $f, g$ and $h$ be the real-valued functions defined on $\mathbb{R}$ as $f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{cases}$,$g(x) = \begin{cases} \frac{\sin(x+1)}{x+1}, & x \neq -1 \\ 1, & x=-1 \end{cases}$ and $h(x) = 2[x] - f(x)$,where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is

JEE MAIN 2023 All JEE MAIN

A
$1$
B
$0$
C
$-1$
D
$2$

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Mix Examples of Relation and Function

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JEE MAIN 2023 Paper All JEE MAIN years →

Let $S = \mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by: $R = \{(x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R}\}$. Then,the sum of all the elements in the range of $R$ is equal to

DifficultJEE MAIN 2025

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Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2e^{2x}}{e^{2x} + e}$. Then $f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \dots + f\left(\frac{99}{100}\right)$ is equal to

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Let $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\cos x, \sin x)$. Define $h(y) = f(x)y^2 + ay + g(x)$. If the equation $h(y) = 0$ has real roots for all $x \in R$,find the complete set of values of $a$.

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Let $f(x) = \begin{cases} x-1, & x \text{ is even} \\ 2x, & x \text{ is odd} \end{cases}$. If for some $a \in N, f(f(f(a))) = 21$,then $\lim_{x \rightarrow a^{-}} \left\{ \frac{|x|^3}{a} - \left[ \frac{x}{a} \right] \right\}$,where $[t]$ denotes the greatest integer less than or equal to $t$,is equal to:

DifficultJEE MAIN 2024

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Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f: A \rightarrow B$ such that $1 \in f(A)$ is equal to:

DifficultJEE MAIN 2025

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Let $S = \mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by: $R = \{(x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R}\}$. Then,the sum of all the elements in the range of $R$ is equal to

Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2e^{2x}}{e^{2x} + e}$. Then $f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \dots + f\left(\frac{99}{100}\right)$ is equal to

Let $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\cos x, \sin x)$. Define $h(y) = f(x)y^2 + ay + g(x)$. If the equation $h(y) = 0$ has real roots for all $x \in R$,find the complete set of values of $a$.

Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f: A \rightarrow B$ such that $1 \in f(A)$ is equal to:

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