Answer
$$32t + {v_0}$$
Work Step by Step
$$\eqalign{
& v = \frac{{32}}{k}\left( {1 - {e^{ - kt}} + \frac{{{v_0}k{e^{ - kt}}}}{{32}}} \right) \cr
& {\text{Rewrite the function}} \cr
& v = \frac{{32\left( {1 - {e^{ - kt}} + \frac{{{v_0}k{e^{ - kt}}}}{{32}}} \right)}}{k} \cr
& v = \frac{{32 - 32{e^{ - kt}} + {v_0}k{e^{ - kt}}}}{k} \cr
& v = \frac{{32\left( {1 - {e^{ - kt}}} \right)}}{k} + {v_0}{e^{ - kt}} \cr
& k{\text{ is approaching zero, then}} \cr
& \mathop {\lim }\limits_{k \to 0} v = \mathop {\lim }\limits_{k \to 0} \frac{{32\left( {1 - {e^{ - kt}}} \right)}}{k} + \mathop {\lim }\limits_{k \to 0} {v_0}{e^{ - kt}} \cr
& {\text{Evaluate the limit}} \cr
& \mathop {\lim }\limits_{k \to 0} v = \mathop {\lim }\limits_{k \to 0} \frac{{32\left( {1 - {e^{ - 0}}} \right)}}{0} + {v_0}\left( 0 \right){e^{ - 0t}} \cr
& {\text{Use L'Hopital's Rule}} \cr
& = \mathop {\lim }\limits_{k \to 0} \frac{{\frac{d}{{dk}}\left[ {32\left( {1 - {e^{ - kt}}} \right)} \right]}}{{\frac{d}{{dk}}\left[ k \right]}} + \mathop {\lim }\limits_{k \to 0} {v_0}{e^{ - kt}} \cr
& = \mathop {\lim }\limits_{k \to 0} \frac{{32\left( {t{e^{ - kt}}} \right)}}{1} + \mathop {\lim }\limits_{k \to 0} {v_0}{e^{ - kt}} \cr
& = \mathop {\lim }\limits_{k \to 0} 32\left( {t{e^{ - kt}}} \right) + \mathop {\lim }\limits_{k \to 0} {v_0}{e^{ - kt}} \cr
& {\text{Evaluate the limit when }}k \to 0 \cr
& = 32\left( {t{e^{ - \left( 0 \right)t}}} \right) + {v_0}{e^{ - \left( 0 \right)t}} \cr
& = 32\left( t \right) + {v_0}\left( 1 \right) \cr
& = 32t + {v_0} \cr} $$
