If
\[
f(x)=
\frac{\lambda e^{\frac1x}+3e^{-\frac1x}}
{(\lambda+2)e^{\frac1x}-e^{-\frac1x}},
\qquad x\neq0
\]
and \(f(0)=k\), \(k\in\mathbb R\), is a continuous function at \(x=0\), then \(2\lambda=\) ?
Show Hint
When exponential terms such as \(e^{1/x}\) and \(e^{-1/x}\) occur, always evaluate left-hand and right-hand limits separately.