Sure! To solve the limit problem [tex]\(\lim_{x \rightarrow 3} \frac{x^2 - 5x + 6}{x - 3}\)[/tex], let's go through the steps in detail:
1. Identify the Indeterminate Form:
First, substitute [tex]\( x = 3 \)[/tex] directly into the given expression:
[tex]\[
\frac{3^2 - 5(3) + 6}{3 - 3} = \frac{9 - 15 + 6}{0} = \frac{0}{0}.
\][/tex]
This yields the indeterminate form [tex]\(\frac{0}{0}\)[/tex], so we'll need to simplify the expression.
2. Factorizing the Numerator:
We factor the quadratic expression [tex]\( x^2 - 5x + 6 \)[/tex]:
[tex]\[
x^2 - 5x + 6 = (x - 2)(x - 3).
\][/tex]
Thus, our limit expression becomes:
[tex]\[
\lim_{x \rightarrow 3} \frac{(x - 2)(x - 3)}{x - 3}.
\][/tex]
3. Simplify the Expression:
We can cancel the common term [tex]\((x - 3)\)[/tex] in the numerator and the denominator:
[tex]\[
\lim_{x \rightarrow 3} \frac{(x - 2)(x - 3)}{x - 3} = \lim_{x \rightarrow 3} (x - 2).
\][/tex]
This simplification is valid for all [tex]\( x \neq 3 \)[/tex].
4. Evaluate the Simplified Expression:
Now, we can find the limit by direct substitution:
[tex]\[
\lim_{x \rightarrow 3} (x - 2) = 3 - 2 = 1.
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \rightarrow 3} \frac{x^2 - 5x + 6}{x - 3} = 1.
\][/tex]
