Question

question 7
evaluate the limit:
\\(\lim_{x \to 7} \frac{x^2 - 2x - 35}{3x - 21}\\) =
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question 8
evaluate the limit

Explanation:

Step1: Check direct substitution

Substitute $x=7$ into the expression:
$\frac{7^2 - 2(7) - 35}{3(7) - 21} = \frac{49 - 14 - 35}{21 - 21} = \frac{0}{0}$
This is an indeterminate form, so we factor.

Step2: Factor numerator and denominator

Factor the quadratic numerator:
$x^2 - 2x - 35 = (x - 7)(x + 5)$
Factor the linear denominator:
$3x - 21 = 3(x - 7)$
Rewrite the limit:
$\lim_{x \to 7} \frac{(x - 7)(x + 5)}{3(x - 7)}$

Step3: Cancel common factors

Cancel $(x - 7)$ (valid since $x
eq 7$ when taking the limit):
$\lim_{x \to 7} \frac{x + 5}{3}$

Step4: Substitute $x=7$

$\frac{7 + 5}{3} = \frac{12}{3}$

Answer:

$4$