Question

limit laws and continuity: problem 2 (1 point) given \\(\lim_{x \to 4} f(x) = 8\\) and \\(\lim_{x \to 4} g(x) = 7\\), evaluate \\(\lim_{x \to 4} \frac{f(x) + g(x)}{5f(x)}\\). (if the limit does not exist, enter \dne\.) limit = \\(\square\\)

Explanation:

Step1: Apply Sum and Quotient Limit Laws

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 non - zero). So, \(\lim_{x
ightarrow4}\frac{f(x)+g(x)}{5f(x)}=\frac{\lim_{x
ightarrow4}(f(x)+g(x))}{\lim_{x
ightarrow4}(5f(x))}\)

Step2: Apply Sum and Constant Multiple Laws

For the numerator, \(\lim_{x
ightarrow4}(f(x)+g(x))=\lim_{x
ightarrow4}f(x)+\lim_{x
ightarrow4}g(x)\) (by sum law). For the denominator, \(\lim_{x
ightarrow4}(5f(x)) = 5\lim_{x
ightarrow4}f(x)\) (by constant multiple law).
We know that \(\lim_{x
ightarrow4}f(x) = 8\) and \(\lim_{x
ightarrow4}g(x)=7\). Substitute these values into the expressions:
Numerator: \(\lim_{x
ightarrow4}f(x)+\lim_{x
ightarrow4}g(x)=8 + 7=15\)
Denominator: \(5\lim_{x
ightarrow4}f(x)=5\times8 = 40\)

Step3: Simplify the Fraction

Now, we have \(\frac{15}{40}\), which simplifies to \(\frac{3}{8}\)

Answer:

\(\frac{3}{8}\)