suppose we want to find lim(x→4) (x² + 3x - 28)/(x² - 11x + 28). if x = 4, the numerator is 0 and the…

suppose we want to find lim(x→4) (x² + 3x - 28)/(x² - 11x + 28). if x = 4, the numerator is 0 and the denominator is 0. this means the limit is 0 the question is wrong the limit is undefined an algebraic reduction is possible. factoring: x² + 3x - 28=(x + 7)(x - 4) x² - 11x + 28=(x - 4)(x - 7) so lim(x→4) (x² + 3x - 28)/(x² - 11x + 28)=lim(x→4) ((x + 7)(x - 4))/((x - 4)(x - 7)) simplifying the rational expression: =lim(x→4)

suppose we want to find lim(x→4) (x² + 3x - 28)/(x² - 11x + 28). if x = 4, the numerator is 0 and the denominator is 0. this means the limit is 0 the question is wrong the limit is undefined an algebraic reduction is possible. factoring: x² + 3x - 28=(x + 7)(x - 4) x² - 11x + 28=(x - 4)(x - 7) so lim(x→4) (x² + 3x - 28)/(x² - 11x + 28)=lim(x→4) ((x + 7)(x - 4))/((x - 4)(x - 7)) simplifying the rational expression: =lim(x→4)

Answer

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+3x - 28=(x + 7)(x - 4)$ and the denominator $x^{2}-11x + 28=(x - 4)(x - 7)$.

Step2: Simplify the rational - expression

Cancel out the common factor $(x - 4)$ in the numerator and denominator. So, $\lim_{x\rightarrow4}\frac{(x + 7)(x - 4)}{(x - 4)(x - 7)}=\lim_{x\rightarrow4}\frac{x + 7}{x - 7}$.

Step3: Evaluate the limit

Substitute $x = 4$ into $\frac{x + 7}{x - 7}$. We get $\frac{4+7}{4 - 7}=\frac{11}{-3}=-\frac{11}{3}$.

Answer:

$\lim_{x\rightarrow4}\frac{x^{2}+3x - 28}{x^{2}-11x + 28}=-\frac{11}{3}$