lim x tending to infinity ((2-x)^40*(4+x)^5)/(2-x)^45
Much the same. IF you were to multiply everything out, the numerator would be x^{45} (NOT -x^45) plus terms involving lower powers of x and the denominator would be -x^{45} plus terms involving lower powers of x.
General rule: If a rational function has polynomials of the same degree in numerator and denominator, the limit,as x goes infinity, is the ratio of those leading coefficients. If the numerator has lower degree than the denominator, the imit, as x goes to infinity, is 0. If the numerator has higher degree than the denominator, the limit, as x goes to infinity, does not exist.