The limit doesn't exist. Notice that [tex]\sqrt{t^2-9}[/tex] is defined only for [tex]t^2\ge9[/tex], or [tex]|t|\ge3[/tex], or [tex]t\le-3[/tex] and [tex]t\ge3[/tex]. In other words, [tex]\dfrac{\sqrt{t^2-9}-3}{t^2}[/tex] doesn't exist for any [tex]t[/tex] between -3 and 3.
