How do I find the limit of a polynomial function?
1 Answer
See the explanation section.
Explanation:
For any polynomial function,
That is
The proof uses the properties of limits.
Every polynomial function (with real coefficients) has from:
where the
= 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)