Question

find the limit, if it exists. (if an answer does not exist, enter dne.)
\\( \lim _ { x \
ightarrow \infty } \left( 3 + \frac { 8 } { x } \
ight) \\)

Explanation:

Step1: Apply limit laws

We know that \(\lim_{x
ightarrow\infty}(a + b)=\lim_{x
ightarrow\infty}a+\lim_{x
ightarrow\infty}b\). So \(\lim_{x
ightarrow\infty}(3+\frac{8}{x})=\lim_{x
ightarrow\infty}3+\lim_{x
ightarrow\infty}\frac{8}{x}\).

Step2: Evaluate \(\lim_{x

ightarrow\infty}3\)
The limit of a constant \(c\) as \(x
ightarrow\infty\) is \(c\). So \(\lim_{x
ightarrow\infty}3 = 3\).

Step3: Evaluate \(\lim_{x

ightarrow\infty}\frac{8}{x}\)
As \(x
ightarrow\infty\), \(\frac{8}{x}
ightarrow0\). Mathematically, \(\lim_{x
ightarrow\infty}\frac{8}{x}=8\lim_{x
ightarrow\infty}\frac{1}{x}\). Since \(\lim_{x
ightarrow\infty}\frac{1}{x}=0\), then \(8\lim_{x
ightarrow\infty}\frac{1}{x}=0\).

Step4: Sum the results

\(\lim_{x
ightarrow\infty}(3+\frac{8}{x})=\lim_{x
ightarrow\infty}3+\lim_{x
ightarrow\infty}\frac{8}{x}=3 + 0\).

Answer:

\(3\)