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)#