$show \ that \ the \ sequence :$
$x_1=3 \ , \ x_{n+1}= \frac{x_n}{2}+ \frac{4}{x_n}$
$is \ convergent \ and \ find \ its \ limit?$
Besides $x_n$ is a subsequence of $x_{n+1},$ so both sequences converge to the same limit (which you need to prove that exists, so call it $l$), then the limit is $l=\frac l2+\frac4l.$