If the function f(x)=gathered (2)/(x) (k_1+1)x+ (k_2-1)x,, x0 gathered is continuous at x=…

Question

If the function

$f(x)=\begin{gathered} \frac{2}{x}{\sin ({k}_{1}+1)x+\sin ({k}_{2}-1)x},\, x0 \end{gathered}$

is continuous at $x=0$, then ${k}_{2}^{1}+{k}_{2}^{2}$ is equal to :

Accepted Answer

$\begin{gathered} \underset{x	o {0}^{-}}{lim}\frac{2}{x}{\sin ({k}_{1}+1)x+\sin ({k}_{2}-1)x}=4 \ \Rightarrow 2({k}_{1}+1)+2({k}_{2}-1)=4 \ \Rightarrow {k}_{1}+{k}_{2}=2 \ \Rightarrow \underset{x	o {0}^{+}}{lim}\frac{2}{x}\ln (\frac{2+{k}_{1}x}{2+{k}_{2}x})=4 \ \Rightarrow \underset{x	o {0}^{+}}{lim}\frac{1}{x}\ln (1+\frac{({k}_{1}-{k}_{2})x}{2+{k}_{2}x})=2 \ \Rightarrow \frac{{k}_{1}-{k}_{2}}{2}=2 \ \Rightarrow {k}_{1}-{k}_{2}=4 \ 	herefore {k}_{1}=3,{k}_{2}=-1 \ {k}_{2}^{1}+{k}_{2}^{2}=9+1=10 \end{gathered}$

Solutions · 1