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

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Jim H Share
Jul 15, 2015

Answer:

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#

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Jul 15, 2015

Answer:

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 = -∞#

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