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  • <img alt="\mathrm{ Let f(x) is\, \, a \, \, function \, \, continuous \, for\, all\, \, , , x \in R except at x=0, such\, \, that\, \, f^{\prime}(x)<0 \forall x \in(-\infty, 0) and }" src="https

\mathrm{ Let f(x) is\, \, a \, \, function \, \, continuous \, for\, all\, \, , , x \in R except at x=0, such\, \, that\, \, f^{\prime}(x)<0 \forall x \in(-\infty, 0) and }\mathrm{ f^{\prime}(x)>0 \forall x \in(0, \infty). Let \lim _{x \rightarrow 0^{+}} f(x)=2, \lim _{x \rightarrow 0^{-}} f(x)=3 \, \, and \, \, f(0)=4. }Find the value of ' a ' for which
2\left(\lim _{x \rightarrow 0} f\left(x^3-x^2\right)\right)=a\left(\lim _{x \rightarrow 0} f\left(2 x^4-x^5\right)\right) .

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

\mathrm{ \begin{aligned} \text { } x \rightarrow 0 ; x^3-x^2 & =x^2(x-1) \rightarrow 0^{-} \\ x \rightarrow 0 ; 2 x^4-x^5 & =x^4(2-x) \rightarrow 0^{+} \\ f\left(0^{-}\right) & =3, f\left(0^{-}\right)=2 \\ 2 \cdot(3) & =a(2) \Rightarrow a=3 . \end{aligned}}

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Pankaj Sanodiya

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