If p is a polynomial show that lim x→ap (x) = p (a

Let p (x) be a polynomial, and suppose that a is any real number. Prove that

lim x→a p (x) = p (a).

Solution. Notice that

2 (-1) 4 - 3 (-1) 3 - 4 (-1) 2 - (-1) - 1 = 1.

So x - (-1) must divide 2x^4 - 3x^3 - 4x^2 - x - 2. Do polynomial long division to get 2x^4 - 3x^3 - 4x^2 - x - 2 / (x - (-1)) = 2x^3 - 5x^2 + x - 2.

Let ε > 0. Set δ = min{ ε/40, 1}. Let x be a real number such that 0 < |x - (-1) | < δ. Then |x + 1| < ε/40. Also, |x + 1| < 1, so - 2 < x < 0. In particular |x| < 2. So

|2x^3 - 5x^2 + x - 2| ≤ |2x^3 | + | - 5x^2 | + |x| + | - 2|

= 2|x|^3 + 5|x|^2 + |x| + 2

< 2 (2) ^3 + 5 (2) ^2 + (2) + 2

= 40

Thus, |2x^4 - 3x^3 - 4x^2 - x - 2| = |x + 1| · |2x^3 - 5x^2 + x - 2| < ε/40 · 40 = ε.