Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 2 / a

Write $R(x)=(x^2+3)(2x^2-1)/(3x^4-x^3-1)$. This rational function is continuous at both finite points because its denominator is nonzero there. Substitution gives the first two limits; at infinity divide numerator and denominator by $x^4$ and use their leading coefficients. Therefore

For fixed real $a$, the base is positive for all sufficiently large positive $x$. Take its logarithm and use the Taylor expansion $\ln (1-u)=-u+O(u^2)$:Continuity of the exponential function then gives $\boxed{\underset{x\to \infty }{\lim }(1-a/x{)}^x=e^{-a}}$. This includes $a=0$.