Question

evaluate the limit: \\(\lim\limits_{x\to 2} \frac{x^2 - 7x + 10}{x - 2}\\) question help: video

Explanation:

Step1: Factor the numerator

We factor the quadratic expression \(x^2 - 7x + 10\). We need two numbers that multiply to \(10\) and add up to \(-7\). Those numbers are \(-2\) and \(-5\). So, \(x^2 - 7x + 10=(x - 2)(x - 5)\).
The limit becomes \(\lim_{x
ightarrow 2}\frac{(x - 2)(x - 5)}{x - 2}\).

Step2: Cancel the common factor

Since \(x
ightarrow 2\) but \(x
eq2\) (we are taking the limit as \(x\) approaches \(2\), not evaluating at \(x = 2\)), we can cancel the common factor \((x - 2)\) in the numerator and the denominator.
After canceling, we get \(\lim_{x
ightarrow 2}(x - 5)\).

Step3: Evaluate the limit

Now we can directly substitute \(x = 2\) into the expression \(x - 5\).
\(2-5=-3\).

Answer:

\(-3\)