lim x → ∞ 16x4 − x3 + x + 4/ 4x4 + 2x3 + x2 + 5x + 2
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it tends to be best to divide it all by the highest power of x that is present
there are 3 general rules for polynomial quotients that can be memorized as well, if you are up to remembering stuff
In case you didn't know all the numbers next to the "X" is a power, ex: 16x^4-x^3 ++4/4x^4+2x^3+x^2+5x+2
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and, what do you get when you divide all the terms by the highest power of x?
(16x4 − x3 + x + 4)/x4
(4x4 + 2x3 + x2 + 5x + 2)/x4
16 +(− x3 + x + 4)/x4
4 + (2x3 + x2 + 5x + 2)/x4
the 'lesser powers' do not overcome the x4 so they will retain an x under them.
anything with an x under it goes to 0 as x to infinity
what does this leave us?
16 + 0
4 + 0
the 3 rules are:
if the degree of the top is bigger than the bottom, the limit to infinity DNE
if the degree of the top is smaller than the bottom, the limit to infinity is 0
if the degree of the top is the same as the bottom, the limit to infinity is the ratio of leading coeffs