Question

limit laws and continuity: problem 2 (1 point) given \\( \lim_{x \to 3} f(x) = 7 \\) and \\( \lim_{x \to 3} g(x) = 8 \\), evaluate \\( \lim_{x \to 3} \frac{f(x) + g(x)}{5f(x)} \\). (if the limit does not exist, enter dne.) limit = \\( \square \\)

Explanation:

Step1: Apply Sum Law and Quotient Law

The limit of a sum is the sum of the limits, and the limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero). So we can rewrite the limit as $\frac{\lim_{x \to 3} f(x) + \lim_{x \to 3} g(x)}{5\lim_{x \to 3} f(x)}$.

Step2: Substitute the given limits

We know that $\lim_{x \to 3} f(x) = 7$ and $\lim_{x \to 3} g(x) = 8$. Substitute these values into the expression: $\frac{7 + 8}{5\times7}$.

Step3: Simplify the numerator and denominator

First, calculate the numerator: $7 + 8 = 15$. Then calculate the denominator: $5\times7 = 35$. So the expression becomes $\frac{15}{35}$.

Step4: Reduce the fraction

Simplify $\frac{15}{35}$ by dividing both the numerator and denominator by their greatest common divisor, which is 5. So $\frac{15\div5}{35\div5} = \frac{3}{7}$.

Answer:

$\frac{3}{7}$