a and b are integers that satisfy: \displaystyle \lim_{x \to 1} \frac{x-1}{x^2-ax+b} = -\frac{1}{3}. What is the value of a+b?
To find the value of a+b, we need to analyze the given limit.
Let's break down the limit and solve it step by step.
\displaystyle \lim_{x \to 1} \frac{x-1}{x^2-ax+b} = -\frac{1}{3}
First, let's factor the denominator by finding the quadratic roots:
x^2 - ax + b = (x - r_1)(x - r_2)
Since we don't have any specific values for a and b, we'll use the quadratic formula to solve for the roots, r_1 and r_2:
r_1,2 = \frac{-(-a) \pm \sqrt{(-a)^2 - 4(1)(b)}}{2(1)}
= \frac{a \pm \sqrt{a^2 - 4b}}{2}
Now, the limit becomes:
\displaystyle \lim_{x \to 1} \frac{x-1}{(x - r_1)(x - r_2)}
To simplify the limit further, we can use partial fraction decomposition. This technique allows us to write a rational function as the sum of simpler fractions.
Using partial fraction decomposition, we can express the rational function as:
\frac{x-1}{(x - r_1)(x - r_2)} = \frac{A}{(x - r_1)} + \frac{B}{(x - r_2)}
Multiplying both sides by (x - r_1)(x - r_2), we get:
x - 1 = A(x - r_2) + B(x - r_1)
Now, we need to find suitable values for A and B to solve the above equation.
Setting x = r_1 gives us:
r_1 - 1 = A(r_1 - r_2)
A = \frac{r_1 - 1}{r_1 - r_2}
Setting x = r_2, we have:
r_2 - 1 = B(r_2 - r_1)
B = \frac{r_2 - 1}{r_2 - r_1}
Now, we rewrite the limit:
\displaystyle \lim_{x \to 1} \left(\frac{A}{(x - r_1)} + \frac{B}{(x - r_2)}\right)
Using the limit properties, we can evaluate the limit separately for each fraction:
\displaystyle \lim_{x \to 1} \frac{A}{(x - r_1)} + \displaystyle \lim_{x \to 1} \frac{B}{(x - r_2)}
Since the limit of the sum is equal to the sum of the limits, we have:
A + B = -\frac{1}{3}
Substituting the values of A and B, we get:
\frac{r_1 - 1}{r_1 - r_2} + \frac{r_2 - 1}{r_2 - r_1} = -\frac{1}{3}
To find a+b, we need to substitute r_1 and r_2 into the expression a+b = \frac{a + \sqrt{a^2 - 4b}}{2} + \frac{a - \sqrt{a^2 - 4b}}{2}.
However, the specific values of a and b are not provided in the question. Without specific values, it is not possible to determine the exact value of a+b. So, the answer to the question remains undetermined.