evaluate the limit. if the limit does not exist, enter dne. lim(x→3) (2x² - 3x - 9)/(x - 3)
evaluate the limit. if the limit does not exist, enter dne. lim(x→3) (2x² - 3x - 9)/(x - 3)
Answer
Explanation:
Step1: Factor the numerator
Factor $2x^{2}-3x - 9$. We get $2x^{2}-3x - 9=2x^{2}-6x + 3x-9=2x(x - 3)+3(x - 3)=(2x + 3)(x - 3)$.
Step2: Simplify the function
The original limit $\lim_{x\rightarrow3}\frac{2x^{2}-3x - 9}{x - 3}$ becomes $\lim_{x\rightarrow3}\frac{(2x + 3)(x - 3)}{x - 3}$. Cancel out the common factor $(x - 3)$ (for $x\neq3$), and we have $\lim_{x\rightarrow3}(2x+3)$.
Step3: Evaluate the limit
Substitute $x = 3$ into $2x+3$. We get $2\times3+3=6 + 3=9$.
Answer:
$9$