(A) To find $\lim_{x \to 0} g(f(x))$,we examine the behavior of $f(x)$ as $x \to 0$.As $x \to 0$ and $x \neq 0$,$f(x) = \sin x$.As $x \to 0$,$f(x) = \

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(A) To find $\lim_{x \to 0} g(f(x))$,we examine the behavior of $f(x)$ as $x \to 0$.As $x \to 0$ and $x \neq 0$,$f(x) = \sin x$.As $x \to 0$,$f(x) = \sin x \to 0$.Let $u = f(x)$. As $x \to 0$,$u \to 0$.In the limit $\lim_{u \to 0} g(u)$,we consider values of $u$ near $0$ but $u \neq 0$.For $u \neq 0$ and $u \neq 2$,$g(u) = u^2 + 1$.Therefore,$\lim_{u \to 0} g(u) = \lim_{u \to 0} (u^2 + 1) = 0^2 + 1 = 1$.Thus,$\lim_{x \to 0} g(f(x)) = 1$.

Class 12 Mathematics Relations and Functions Composition of Functions

Easy

If $f(x) = \begin{cases} \sin x, & x \neq n\pi, n \in \mathbb{Z} \\ 0, & \text{otherwise} \end{cases}$ and $g(x) = \begin{cases} x^2 + 1, & x \neq 0, 2 \\ 4, & x = 0 \\ 5, & x = 2 \end{cases}$,then $\lim_{x \to 0} g(f(x)) = $

A
$1$
B
$0$
C
$\frac{1}{2}$
D
$\frac{1}{4}$

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Class 12

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Relations and Functions

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Composition of Functions

Consider functions $f$ and $g$ such that the composite function $g \circ f$ is defined and is one-one. Are $f$ and $g$ both necessarily one-one?

Easy

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If $R$ is a relation $ < $ from $A = \{1, 2, 3, 4\}$ to $B = \{1, 3, 5\}$ i.e.,$(a, b) \in R \iff a < b$,then $R \circ R^{-1}$ is

Easy

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If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) = x^{2} - 3x + 4$ and $g(x) = 2x + 1$,then the value of $x$ for which $f(x) = (f \circ g)(x)$ is

EasyMHT CET 2020

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If $f$ and $g$ are two increasing functions such that $fog$ is defined,then what kind of function is $fog$?

Medium

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If $f(x)=e^x$ and $g(x)=\ln(x)$ for all $x \in [1, \infty)$,then $f \circ g$ is . . . . . .

EasyAP EAMCET 2020

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If $f(x) = \begin{cases} \sin x, & x \neq n\pi, n \in \mathbb{Z} \\ 0, & \text{otherwise} \end{cases}$ and $g(x) = \begin{cases} x^2 + 1, & x \neq 0, 2 \\ 4, & x = 0 \\ 5, & x = 2 \end{cases}$,then $\lim_{x \to 0} g(f(x)) = $

Similar Questions

Consider functions $f$ and $g$ such that the composite function $g \circ f$ is defined and is one-one. Are $f$ and $g$ both necessarily one-one?

If $R$ is a relation $ < $ from $A = \{1, 2, 3, 4\}$ to $B = \{1, 3, 5\}$ i.e.,$(a, b) \in R \iff a < b$,then $R \circ R^{-1}$ is

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) = x^{2} - 3x + 4$ and $g(x) = 2x + 1$,then the value of $x$ for which $f(x) = (f \circ g)(x)$ is

If $f$ and $g$ are two increasing functions such that $fog$ is defined,then what kind of function is $fog$?

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