Generally speaking, to find , where f and g are polynomials, first try setting x= a.
If g(a) is not 0, then the limit is just f(a)/g(a).
If g(a)= 0 and f(a) is not 0, then the limit does not exist.
If both f(a)= 0 and g(a)= 0, then, yes, x- a must be a factor of both. Just divide the polynomials by x-a to find the other factor and cancel the x-a in numerator and denominator.
For (c), "rationalize the denominator" by multiplying both numerator and denominator by
For (d), use the fact that has limit 1 as a goes to 0. (Of course, "a", here, is "5x".)