A and b are fixed real numbers. evaluate the limit as (t) approaches (a) ((sqrt(3t+1)-sqrt(3a+1))/(t-a))

Use the conjugate.

lim(x→a) [√(3t+1) - √(3a+1)] / (t - a)
= lim(x→a) [√(3t+1) - √(3a+1)] * [√(3t+1) + √(3a+1)] / {(t - a) [√(3t+1) + √(3a+1)]}
= lim(x→a) [(3t+1) - (3a+1)] / {(t - a) [√(3t+1) + √(3a+1)]}
= lim(x→a) 3(t - a) / {(t - a) [√(3t+1) + √(3a+1)]}
= lim(x→a) 3 / [√(3t+1) + √(3a+1)]
= 3 / [√(3a+1) + √(3a+1)]
= 3 / [2√(3a+1)].

I hope this helps!