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!
Step-by-step explanation:
do 51,416×4
ans is 245,664
Answer:
Correct
Step-by-step explanation:
When you type 1/3 into a calculator it should give you something along the lines of .333333334. If you continue to round, eventually you will get 1.
Answer:
It has no amplitud and the period is 5pi/2
Step-by-step explanation:
Given a function of the following type:
f(t) = AtanB(t + C)
The function has no amplitud, given that it doesn't have maximum or minimum value. And the period is given by: pi/B
In this case, we have f(t)= -tan0.4t. Then:
B = 0.4
⇒ Period = pi/0.4 = 5pi/2
Therefore, the answer is: It has no amplitud and the period is 5pi/2
Answer:
positive
negative
negative
Step-by-step explanation:
