Q.Evaluate $\lim_{x \to \frac{1}{2}} \left( \dfrac{8x - 3}{2x - 1} - \dfrac{4x^2 + 1}{4x^2 - 1} \right)$.

Once the limits of two functions f and g at a point a are known — say f(x) tends to l and g(x) tends to m as x tends to a — the algebra of limits lets a complicated limit be broken apart into the limits of its simpler pieces rather than evaluated from scratch. The sum or difference of the functions tends to l plus or minus m; the product tends to l times m; a constant multiple k times f(x) tends to kl; and the quotient f(x)/g(x) tends to l/m, provided the denominator's limit m is not zero. Standing building blocks that follow from these rules include: the limit of a constant is itself, the limit of x is a, the limit of x to a natural-number power n is a^n, and for any polynomial p(x), the limit as x tends to a is simply p(a) — …