Let f: R-0 R be a function such that f(x)-6 f((1)/(x))=(35)/(3 x)-(5)/(2).

Question

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}$. If the $\lim\limits _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in \mathbb{R}$, then $\alpha+2 \beta$ is equal to

Accepted Answer

$F(x)-6 f(1 / x)=\frac{35}{3 x}-\frac{5}{2}\quad\text{..... (1)}$

$$\begin{aligned}
&\begin{aligned}
& \text { Replace } x \rightarrow \frac{1}{x} \
& F(1 / x)-6(x)=\frac{35 x}{3}-\frac{5}{2}\quad\text{..... (2)}
\end{aligned}\
\&\text { Using (1) \& (2) }\
\&\begin{aligned}
& f(x)=-2 x-\frac{1}{3 x}+\frac{1}{2} \
& B=\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)
\end{aligned}
\end{aligned}$$

$$\begin{aligned}
&= \lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}-2 x-\frac{1}{3 x}+\frac{1}{2}\right) \
& \alpha=3, \quad B=1 / 2 \
& \text { So, } \alpha+2 B=3+1=4
\end{aligned}$$

Solutions · 1