determine the limit shown below in simplest form.\n lim_{x\rightarrow5}\frac{x^{2}-13x + 40}{x^{2}-8x + 15}

Answer:

$\frac{3}{2}$

Explanation:

Step1: Factor the numerator and denominator

$x^{2}-13x + 40=(x - 5)(x - 8)$ and $x^{2}-8x + 15=(x - 5)(x - 3)$ So, $\lim_{x\rightarrow5}\frac{x^{2}-13x + 40}{x^{2}-8x + 15}=\lim_{x\rightarrow5}\frac{(x - 5)(x - 8)}{(x - 5)(x - 3)}$

Step2: Cancel out the common factor

Since $x\neq5$ when taking the limit, we can cancel out the $(x - 5)$ terms. We get $\lim_{x\rightarrow5}\frac{x - 8}{x - 3}$

Step3: Substitute $x = 5$

Substitute $x = 5$ into $\frac{x - 8}{x - 3}$, we have $\frac{5-8}{5 - 3}=\frac{-3}{2}=-\frac{3}{2}$