# How do I find the limit of a polynomial function?

Oct 14, 2015

See the explanation section.

#### Explanation:

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

That is ${\lim}_{x \rightarrow a} P \left(x\right) = P \left(a\right)$.

The proof uses the properties of limits.

Every polynomial function (with real coefficients) has from:

$P \left(x\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \cdot \cdot \cdot + {a}_{1} x + {a}_{0}$

where the ${a}_{i}$ are real numbers and $n$ is a nonnegative integer.

${\lim}_{x \rightarrow a} P \left(x\right) = {\lim}_{x \rightarrow a} \left[{a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \cdot \cdot \cdot + {a}_{1} x + {a}_{0}\right]$

$= {\lim}_{x \rightarrow a} \left[{a}_{n} {x}^{n}\right] + {\lim}_{x \rightarrow a} \left[{a}_{n - 1} {x}^{n - 1}\right] + \cdot \cdot \cdot + {\lim}_{x \rightarrow a} \left[{a}_{1} x\right] + {\lim}_{x \rightarrow a} \left[{a}_{0}\right]$ (sum property of limits)

$= {a}_{n} {\lim}_{x \rightarrow a} \left[{x}^{n}\right] + {a}_{n - 1} {\lim}_{x \rightarrow a} \left[{x}^{n - 1}\right] + \cdot \cdot \cdot + {a}_{1} {\lim}_{x \rightarrow a} \left[x\right] + {a}_{0} {\lim}_{x \rightarrow a} \left[1\right]$ (constant multiple rule)

$= {a}_{n} {\left({\lim}_{x \rightarrow a} x\right)}^{n} + {a}_{n - 1} {\left({\lim}_{x \rightarrow a} x\right)}^{n - 1} + \cdot \cdot \cdot + {a}_{1} \left({\lim}_{x \rightarrow a} x\right) + {a}_{0} {\lim}_{x \rightarrow a} \left(1\right)$ (integer power rule or repeated applications of the product rule)

$= {a}_{n} {a}^{n} + {a}_{n - 1} {a}^{n - 1} + \cdot \cdot \cdot + {a}_{1} a + {a}_{0}$ (limits of identity and constant functions)

$= P \left(a\right)$