Let f be the function defined on {-3, 1} by f (x)= (x+1)/x^2+ax+b and (C) its representative curve in an orthonormal system .
Knowing that the two lines of equations x=1 and x=-3 are asymptotes to (C) calculate the values of a and b.
To find the values of a and b, we can use the fact that the lines x=1 and x=-3 are asymptotes to the curve (C).
First, let's determine the behavior of the function f(x) as x approaches ±∞. Since the lines x=1 and x=-3 are asymptotes, the function f(x) must approach ∞ as x approaches ±∞.
To find the behavior of f(x) as x → ±∞, we take the limit of f(x) as x approaches ±∞.
lim (x→±∞) f(x) = lim (x→±∞) [(x+1)/(x^2+a·x+b)]
Since x increases or decreases without bound, we can ignore the 1 in the numerator. We can also ignore the x^2 term since it becomes insignificant compared to the x or b terms.
lim (x→±∞) f(x) = lim (x→±∞) [1/(a·x+b)]
To approach ∞, the denominator should approach 0. Therefore, the value of b must be 0.
lim (x→±∞) [1/(a·x+b)] = lim (x→±∞) [1/(a·x)]
To approach 0, the denominator should be of higher order than the numerator. That means a should be positive.
Therefore, the value of a is a > 0.
In summary, the values of a and b are a > 0 and b = 0.