Estimate the limit numerically or state that the lim _(xarrow 4)(x^2-2x-8)/(x^2)-3x-4= (Enter Fif the limit doesn't exist)

To estimate the limit numerically, we can substitute values of x that are close to 4 into the expression and observe the behavior of the function.<br /><br />Let's substitute x = 3.9 and x = 4.1 into the expression:<br /><br />$\lim _{x\rightarrow 4}\frac {x^{2}-2x-8}{x^{2}-3x-4} = \frac{(3.9)^{2}-2(3.9)-8}{(3.9)^{2}-3(3.9)-4} = \frac{15.21-7.8-8}{15.21-11.7-4} = \frac{-0.59}{-0.49} \approx 1.20$<br /><br />$\lim _{x\rightarrow 4}\frac {x^{2}-2x-8}{x^{2}-3x-4} = \frac{(4.1)^{2}-2(4.1)-8}{(4.1)^{2}-3(4.1)-4} = \frac{16.81-8.2-8}{16.81-12.3-4} = \frac{0.61}{0.51} \approx 1.20$<br /><br />As we can see, the function approaches the value of 1.20 as x approaches 4. Therefore, the limit exists and is equal to 1.20.