To evaluate the limit [tex]\(\lim_{x \to \infty} (2x - \ln x)\)[/tex], let's analyze the behavior of each term as [tex]\(x\)[/tex] approaches infinity.
1. Behavior of [tex]\(2x\)[/tex] as [tex]\(x \to \infty\)[/tex]:
- The term [tex]\(2x\)[/tex] grows linearly with [tex]\(x\)[/tex]. As [tex]\(x\)[/tex] approaches infinity, [tex]\(2x\)[/tex] will also approach infinity. Mathematically, [tex]\(\lim_{x \to \infty} 2x = \infty\)[/tex].
2. Behavior of [tex]\(\ln x\)[/tex] as [tex]\(x \to \infty\)[/tex]:
- The natural logarithm function [tex]\(\ln x\)[/tex] grows much more slowly compared to linear functions. As [tex]\(x\)[/tex] approaches infinity, [tex]\(\ln x\)[/tex] will also approach infinity, but at a much slower rate. Mathematically, [tex]\(\lim_{x \to \infty} \ln x = \infty\)[/tex].
Now, we need to consider the combined effect of these two terms, i.e., [tex]\(2x - \ln x\)[/tex], as [tex]\(x\)[/tex] approaches infinity.
- Since [tex]\(2x\)[/tex] grows much faster than [tex]\(\ln x\)[/tex], the difference [tex]\(2x - \ln x\)[/tex] will be dominated by the term [tex]\(2x\)[/tex].
- As [tex]\(x\)[/tex] becomes very large, the contribution of [tex]\(\ln x\)[/tex] becomes negligible compared to the rapidly increasing [tex]\(2x\)[/tex].
Therefore, the limit of the expression [tex]\(2x - \ln x\)[/tex] as [tex]\(x \to \infty\)[/tex] is determined primarily by the behavior of [tex]\(2x\)[/tex].
Putting this together, we get:
[tex]\[
\lim_{x \to \infty} (2x - \ln x) = \infty
\][/tex]
So, the limit is [tex]\(\infty\)[/tex].
