Question

question
determine the limit shown below in simplest form.

$$ lim _ { x ightarrow 4 } \frac { x ^ { 2 } - 7 x } { x ^ { 2 } - 4 x } $$

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - 7x=x(x - 7)\) and \(x^2-4x=x(x - 4)\). So the expression becomes \(\lim_{x
ightarrow4}\frac{x(x - 7)}{x(x - 4)}\).

Step2: Simplify the expression

Cancel out the common factor \(x\) (for \(x
eq0\), and as \(x
ightarrow4\), \(x
eq0\)). The expression simplifies to \(\lim_{x
ightarrow4}\frac{x - 7}{x - 4}\).

Step3: Substitute \(x = 4\)

Substitute \(x = 4\) into \(\frac{x - 7}{x - 4}\). We get \(\frac{4-7}{4 - 4}=\frac{-3}{0}\). Since the numerator is non - zero (\(-3
eq0\)) and the denominator approaches \(0\) as \(x
ightarrow4\), the limit does not exist.

Answer:

The limit \(\lim_{x
ightarrow4}\frac{x^2 - 7x}{x^2 - 4x}\) does not exist.