Question

question 4
evaluate the limit: \\(\lim_{x \to -4} \frac{x^2 + 7x + 12}{x + 4}\\)
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question 5
evaluate the limit \\(\lim_{x \to 3} \frac{7x^2 - 7x + 5}{x - 7}\\)

Explanation:

Step1: Factor the numerator

$x^2 + 7x + 12 = (x+3)(x+4)$

Step2: Cancel common factors

$\lim_{x \to -4} \frac{(x+3)(x+4)}{x+4} = \lim_{x \to -4} (x+3)$

Step3: Substitute $x=-4$

$\lim_{x \to -4} (x+3) = -4 + 3$

Step1: Substitute $x=7$ directly

$\lim_{x \to 7} \frac{7x^2 -7x +5}{x-7} = \frac{7(7)^2 -7(7) +5}{7-7}$

Step2: Calculate numerator and denominator

Numerator: $7(49)-49+5=343-49+5=299$; Denominator: $0$

Step3: Analyze the limit behavior

As $x \to 7$, denominator approaches 0, numerator approaches 299 (non-zero).

Answer:

$-1$

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