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:
Area = 168m^2
Step-by-step explanation:
Area of triangle = 1/2×base×height
Area = 1/2bh
= 1/2(8+16)(14)
= 168m^2
Answer:
3628800
Step-by-step explanation:
There are 10 options for the first number.
That leaves 9 options for the second number.
That leaves 8 options for the third number.
So on and so forth.
The number of ways 10 numbers can be arranged is:
10×9×8×7×6×5×4×3×2×1
= 10!
= 3628800
If it can scan 72 photos in 8 minutes, it can scan 72/8 = 9 photos per minute.
So in 23 minutes, it can scan 9x23 = 207 photos
Answer:
x=2
Step-by-step explanation:
4( 2+x) = 18-x
8 + 4x = 18 - x
4x+x = 18-8
5x= 10
x=2
