Let f(x) = p(x)/q(x), where p and q are polynomials and reduced to lowest terms. (If p and q have a common factor, then they contribute removable discontinuities ('holes').)
Write this in cases:
(i) If deg p(x) ≤ deg q(x), then f(x) is a proper rational function, and lim(x→ ±∞) f(x) = constant.
If deg p(x) < deg q(x), then these limits equal 0, thus yielding the horizontal asymptote y = 0.
If deg p(x) = deg q(x), then these limits equal a/b, where a and b are the leading coefficients of p(x) and q(x), respectively. Hence, we have the horizontal asymptote y = a/b.
Note that there are no obliques asymptotes in this case. ------------- (ii) If deg p(x) > deg q(x), then f(x) is an improper rational function.
By long division, we can write f(x) = g(x) + r(x)/q(x), where g(x) and r(x) are polynomials and deg r(x) < deg q(x).
As in (i), note that lim(x→ ±∞) [f(x) - g(x)] = lim(x→ ±∞) r(x)/q(x) = 0. Hence, y = g(x) is an asymptote. (In particular, if deg g(x) = 1, then this is an oblique asymptote.)
This time, note that there are no horizontal asymptotes. ------------------ In summary, the degrees of p(x) and q(x) control which kind of asymptote we have.
I hope this helps!
Answer: =4x+18
Step-by-step explanation:
Let's simplify step-by-step.
4x−7+25
=4x+−7+25
Combine Like Terms:
=4x+−7+25
=(4x)+(−7+25)
=4x+18
Answer:
=4x+18
Answer:
You can regroup it 3 times and a remainder of 43. :) hope it helped!
Step-by-step explanation:
Answer:
the answer is 44. . .
Step-by-step explanation:
20+24 =44
Day 3 is A because it tell how many pieces of candies eaten in a minute.
Day 4 is 10 and 15 because the ratios go in a pattern by 5
Day 5 is D but I really dont know that one
