Question

Sequences & Limits help. *question attached below* Will give medal

Answer 1

So I am so clueless as to how to go about tackling this problem. Can someone explain to me?

Answer 2

Hint : if the limit exists, then \(l=\lim\limits_{n\to\infty} x_n =\lim\limits_{n\to\infty} x_{n+1} \)

Answer 3

\[\large \begin{align}
l&=\lim\limits_{n\to\infty} x_{n+1}\\~\\
&=\lim\limits_{n\to\infty} \frac{1}{2}\left(x_n + \frac{a}{x_n}\right)\\~\\
&=\frac{1}{2}\left(\lim\limits_{n\to\infty} x_n + \lim\limits_{n\to\infty} \frac{a}{x_n}\right)\\~\\
&=\frac{1}{2}\left(l+ \frac{a}{l}\right)\\~\\

\end{align}\]

Answer 4

so we have
\[l = \frac{1}{2}\left(l+ \frac{a}{l}\right)\]
you can solve \(l\)

Answer 5

Got it! Thank you :)