# How do I find the limit as x approaches negative infinity of a polynomial?

Jul 15, 2015

If the polynomial $p \left(x\right)$ is of degree $n$ and the coefficient of the highest degree term ${a}_{n}$ is positive, then

lim_{xto-∞}p(x) =∞ if $n$ is even and -∞ if $n$ is odd.

#### Explanation:

We can use the following fact about polynomials:

If p(x) = a_nx^n + a_(n-1)x^(n-1) + … + a_1x + a_0 is a polynomial of degree $n$, then

lim_{xto∞}p(x) = lim_{xto∞}a_nx^n and

lim_{xto-∞}p(x) = lim_{xto-∞}a_nx^n

This fact is really saying that when we take a limit at infinity for a polynomial, all we have to do is look at the term with the largest power and ask what that term is doing in the limit.

If $a$ is positive and $n$ is even, then ${a}_{n} {x}^{n}$ is always positive, and lim_{xto-∞}p(x) = lim_{xto-∞}a_nx^n = ∞

If $a$ is positive and $n$ is odd, then ${a}_{n} {x}^{n}$ is negative when $x$ is negative. So

lim_{xto-∞}p(x) = lim_{xto-∞}a_nx^n = -∞

Jul 15, 2015

I usually do this informally.

#### Explanation:

I ask myself what kinds of numbers do I get if I put in more and more negative numbers for $x$. (Often described by saying "bigger and bigger negative numbers".) ("Big" means "far from zero".)

Examples:

Example 1
$f \left(x\right) = 3 {x}^{4} - 7 {x}^{3} + 2 x + 72$

For very very big numbers, the only term that matters is the largest power term: $3 {x}^{4}$. As I put in bigger and bigger negatives, do I get bigger and bigger positives or negatives?

${x}^{4}$ will always be positive and when I multiply by $3$ the answer will still be positive, so I get bigger and bigger positive numbers.

${\lim}_{x \rightarrow - \infty} f \left(x\right) = \infty$

Example 2
$g \left(x\right) = 5 {x}^{7} + 43 {x}^{4} + 2 {x}^{3} - 5 x + 21$

For very very big numbers, the only term that matters is the largest power term: $5 {x}^{7}$. As I put in bigger and bigger negatives, do I get bigger and bigger positives or negatives?

${x}^{7}$ will be negative for negative $x$'s and when I multiply by $5$ the answer will still be negative, so I get bigger and bigger negative numbers.

${\lim}_{x \rightarrow - \infty} g \left(x\right) = - \infty$

Example 3 (last)
$h \left(x\right) = - 8 {x}^{6} + 7 x - 3$

For very very big numbers, the only term that matters is the largest power term: $- 8 {x}^{6}$. As I put in bigger and bigger negatives, do I get bigger and bigger positives or negatives?

${x}^{6}$ will be positive for all $x$'s and when I multiply by $- 8$ the answer will become negative, so I get bigger and bigger negative numbers.

${\lim}_{x \rightarrow - \infty} h \left(x\right) = - \infty$