How do I find the limit of a polynomial function?

1 Answer
Oct 14, 2015

See the explanation section.

Explanation:

For any polynomial function, P(x), and for and real number a, we can find the limit as x approaches a, by substitution.

That is lim_(xrarra)P(x) = P(a).

The proof uses the properties of limits.

Every polynomial function (with real coefficients) has from:

P(x) = a_nx^n+a_(n-1)x^(n-1)+ * * * +a_1x+a_0

where the a_i are real numbers and n is a nonnegative integer.

lim_(xrarra)P(x) = lim_(xrarra) [a_nx^n+a_(n-1)x^(n-1)+ * * * +a_1x+a_0]

= lim_(xrarra)[a_nx^n]+lim_(xrarra)[a_(n-1)x^(n-1)]+ * * * +lim_(xrarra)[a_1x]+lim_(xrarra)[a_0] (sum property of limits)

= a_nlim_(xrarra)[x^n]+a_(n-1)lim_(xrarra)[x^(n-1)]+ * * * +a_1lim_(xrarra)[x]+a_0lim_(xrarra)[1] (constant multiple rule)

= a_n(lim_(xrarra)x)^n+a_(n-1)(lim_(xrarra)x)^(n-1)+ * * * +a_1(lim_(xrarra)x)+a_0lim_(xrarra)(1) (integer power rule or repeated applications of the product rule)

= a_na^n+a_(n-1)a^(n-1)+ * * * +a_1a+a_0 (limits of identity and constant functions)

= P(a)