Question

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

Answer 1

If the polynomial `p(x)` 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_nx^n` is always positive, and `lim_{xto-∞}p(x) = lim_{xto-∞}a_nx^n = ∞`

If `a` is positive and `n` is odd, then `a_nx^n` is negative when `x` is negative. So

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

Answer 2

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(x) = 3x^4-7x^3+2x+72`

For very very big numbers, the only term that matters is the largest power term: `3x^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_(xrarr-oo)f(x) = oo`

Example 2
`g(x) = 5x^7+43x^4+2x^3-5x+21`

For very very big numbers, the only term that matters is the largest power term: `5x^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_(xrarr-oo)g(x) = -oo`

Example 3 (last)
`h(x) = -8x^6+7x-3`

For very very big numbers, the only term that matters is the largest power term: `-8x^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_(xrarr-oo)h(x) = -oo`