[tex]\displaystyle\\
\frac{x+1}{x}= 9\\\\
x+1 = 9x\\\\
9x-x=1\\\\
8x=1\\\\
x = \frac{1}{8} \\\\
\frac{x^2+1}{x^2}= \frac{\left(\frac{1}{8} \right)^2+1}{\left(\frac{1}{8} \right)^2}= \frac{\frac{1}{64} +1}{\frac{1}{64}}= \frac{\frac{64+1}{64}}{\frac{1}{64}}= \frac{\frac{65}{64}}{\frac{1}{64}}=\frac{65}{64}\cdot \frac{64}{1}= \frac{65}{1}=\boxed{65}[/tex]