Question

How do I find the limit of a polynomial function?

Answer 1

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)`