suppose we want to find lim (x^2 + 3x - 28)/(x^2 - 11x + 28) as x->4. if x = 4, the numerator is 0 and the…

suppose we want to find lim (x^2 + 3x - 28)/(x^2 - 11x + 28) as x->4. 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: x^2 + 3x - 28=(x + 7)(x - 4) x^2 - 11x + 28=(x - 4)(x - 7) so lim (x^2 + 3x - 28)/(x^2 - 11x + 28) as x->4 = lim ((x + 7)(x - 4))/((x - 4)(x - 7)) as x->4 simplifying the rational expression: = lim as x->4 -11/3 hint: remove any common factors you can find.

suppose we want to find lim (x^2 + 3x - 28)/(x^2 - 11x + 28) as x->4. 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: x^2 + 3x - 28=(x + 7)(x - 4) x^2 - 11x + 28=(x - 4)(x - 7) so lim (x^2 + 3x - 28)/(x^2 - 11x + 28) as x->4 = lim ((x + 7)(x - 4))/((x - 4)(x - 7)) as x->4 simplifying the rational expression: = lim as x->4 -11/3 hint: remove any common factors you can find.

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 - function

We have $\lim_{x\rightarrow4}\frac{x^{2}+3x - 28}{x^{2}-11x + 28}=\lim_{x\rightarrow4}\frac{(x + 7)(x - 4)}{(x - 4)(x - 7)}$. Since $x\neq4$ when taking the limit, we can cancel out the common factor $(x - 4)$. So it becomes $\lim_{x\rightarrow4}\frac{x + 7}{x - 7}$.

Step3: Substitute $x = 4$ into the simplified function

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

Answer:

$-\frac{11}{3}$