Question

consider the function defined by ( f(x)=\frac{x^{3}-2 x^{2}-5 x + 6}{(x + 2)(x - 3)} ).
find the limit of ( f ) at ( x = 3 ).
give an exact answer.

Explanation:

Step1: Factor the numerator and denominator

Given \( f(x)=\frac{x^{3}-2x^{2}-5x + 6}{(x + 2)(x-3)}\).
We use polynomial long - division or synthetic division to factor \(x^{3}-2x^{2}-5x + 6\).
By synthetic division with \(x = 1\): \(1\mid1\quad-2\quad-5\quad6\), \(1\times1 = 1\), \(-2 + 1=-1\), \(1\times(-1)=-1\), \(-5+(-1)=-6\), \(1\times(-6)=-6\), \(6+(-6) = 0\).
So \(x^{3}-2x^{2}-5x + 6=(x - 1)(x^{2}-x - 6)\).
And \(x^{2}-x - 6=(x-3)(x + 2)\).
Then \(f(x)=\frac{(x - 1)(x-3)(x + 2)}{(x + 2)(x-3)}\).

Step2: Simplify the function

Cancel out the common factors \((x + 2)\) and \((x - 3)\) (for \(x
eq-2\) and \(x
eq3\)).
We get \(f(x)=x - 1\) for \(x
eq-2\) and \(x
eq3\).

Step3: Find the limit

We want to find \(\lim_{x
ightarrow3}f(x)\).
Since \(f(x)=x - 1\) for \(x
eq3\) (the function is defined in a deleted neighborhood of \(x = 3\)), we use the direct - substitution method.
Substitute \(x = 3\) into \(y=x - 1\).

Answer:

\(2\)