Question

lim
x→∞
1 + 9/x
x/10

Explanation:

Step1: Analyze the limit of each term

As \(x\to\infty\), for the term \(\frac{9}{x}\), we know that \(\lim_{x\to\infty}\frac{c}{x}=0\) (where \(c = 9\) is a constant). For the term \(\frac{10}{x}\), also \(\lim_{x\to\infty}\frac{10}{x}=0\) since \(\lim_{x\to\infty}\frac{k}{x}=0\) (\(k = 10\) is a constant).

Step2: Calculate the limit of the whole expression

We use the sum - rule of limits \(\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)\). Here \(f(x)=1+\frac{9}{x}\) and \(g(x)=\frac{10}{x}\), and \(a=\infty\).
\(\lim_{x\to\infty}(1 + \frac{9}{x}+\frac{10}{x})=\lim_{x\to\infty}1+\lim_{x\to\infty}\frac{9}{x}+\lim_{x\to\infty}\frac{10}{x}\)
Since \(\lim_{x\to\infty}1 = 1\), \(\lim_{x\to\infty}\frac{9}{x}=0\), and \(\lim_{x\to\infty}\frac{10}{x}=0\)

Answer:

\(1\)