This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the limit of r(t), we need to specify the value t approaches. Based on the previous question where t=1 was excluded from the domain, we will find the limit as t 1. We find the limit of each component function separately: _t 1 r(t) = _t 1 (t^2 - 1)/(t - 1), _t 1 sqrt(t+8), _t 1 (( t))/( t) Step 1: Find the limit of the first component. _t 1 (t^2 - 1)/(t - 1) This is an indeterminate form (0)/(0). We can factor the numerator: _t 1 ((t-1)(t+1))/(t - 1) = _t 1 (t+1) = 1+1 = 2 Step 2: Find the limit of the second component. _t 1 sqrt(t+8) This function is continuous at t=1, so we can substitute the value: _t 1 sqrt(t+8) = sqrt(1+8) = sqrt(9) = 3 Step 3: Find the limit of the third component. _t 1 (( t))/( t) As t 1, the numerator ( t) () = 0, and the denominator t (1) = 0. This is an indeterminate form (0)/(0), so we can use L'Hôpital's Rule. Take the derivative of the numerator and the denominator: Derivative of numerator: (d)/(dt)(( t)) = ( t) Derivative of denominator: (d)/(dt)( t) = (1)/(t) Now, apply L'Hôpital's Rule: _t 1 ( ( t))/(1)t = ( ( · 1))/(1)1 = ( ())/(1) = ( (-1))/(1) = - Step 4: Combine the limits of the components. _t 1 r(t) = 2, 3, - The limit of r(t) as t 1 is: _t 1 r(t) = 2 \\ 3 \\ - Last free one today — make it count tomorrow, or type /upgrade for unlimited.