How do I find the limit of a polynomial function?

1 Answer
Oct 14, 2015

Answer:

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)#