Question

lim_{x\to 8} (x - \ln x) solution: 8 - 8; lim_{x\to 8} (\frac{e^x}{x}); lim_{x\to 8} (e^x - x)

Explanation:

Step1: Analyze the limit of \(x\) and \(\ln x\) as \(x\to\infty\)

As \(x\) approaches infinity, \(x\) grows without bound (\(\lim_{x\to\infty} x = \infty\)) and \(\ln x\) also approaches infinity, but \(x\) grows much faster than \(\ln x\) (since the derivative of \(x\) is \(1\) and the derivative of \(\ln x\) is \(\frac{1}{x}\), which approaches \(0\) as \(x\to\infty\)).

Step2: Evaluate \(\lim_{x\to\infty} (x - \ln x)\)

We can think of the behavior of \(x - \ln x\) as \(x\) becomes very large. Since \(x\) dominates \(\ln x\) in growth rate, as \(x\to\infty\), \(x - \ln x\) will also approach infinity. Mathematically, we can also rewrite it (though not necessary here) or just use the growth rate comparison. Since \(\lim_{x\to\infty} x=\infty\) and \(\lim_{x\to\infty} \ln x=\infty\), but \(x\) grows faster, so \(\lim_{x\to\infty} (x - \ln x)=\infty\).

Answer:

\(\lim_{x\to\infty} (x - \ln x)=\infty\)