ધારો કે f: mathbb{R} rightarrow mathbb{R} એક | vedclass

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(C) $\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right) f(x) = \lim _{x \rightarrow \infty} \frac{\sin (1/x)}{1/x} \cdot f(x) = 1 \cdot M =

Vedclass · Accepted Answer

$\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x)=M$

Vedclass · Answer

(C) $\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right) f(x) = \lim _{x \rightarrow \infty} \frac{\sin (1/x)}{1/x} \cdot f(x) = 1 \cdot M = M$. આ સત્ય છે.$(B)$ $\sin(x)$ એ સતત વિધેય હોવાથી,$\lim _{x \rightarrow \infty} \sin(f(x)) = \sin(\lim _{x \rightarrow \infty} f(x)) = \sin M$. આ સત્ય છે.$(C)$ $\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x) = \lim _{x \rightarrow \infty} (x e^{-x}) \cdot \frac{\sin (e^{-x})}{e^{-x}} \cdot f(x)$. $\lim _{x \rightarrow \infty} x e^{-x} = 0$ હોવાથી,લક્ષ $0 \cdot 1 \cdot M = 0$ થાય. તેથી,લક્ષ $M$ છે તે વિધાન ખોટું છે.$(D)$ $\lim _{x \rightarrow \infty} \frac{\sin x}{x} f(x) = (\lim _{x \rightarrow \infty} \frac{\sin x}{x}) \cdot (\lim _{x \rightarrow \infty} f(x)) = 0 \cdot M = 0$. આ સત્ય છે.

ધારો કે $f: \mathbb{R} \rightarrow \mathbb{R}$ એક વિધેય છે જેથી $\lim _{x \rightarrow \infty} f(x)=M > 0$. તો નીચેનામાંથી કયું ખોટું છે?

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