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Problem 55 Medium Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to \infty} (x - \ln x) $

Answer

The limit has the form $\infty-\infty$ and we will change the form to a product by factoring out $x$
\[
\lim _{x \rightarrow \infty}(x-\ln x)=\lim _{x \rightarrow \infty} x\left(1-\frac{\ln x}{x}\right)=\infty \text { since } \lim _{x \rightarrow \infty} \frac{\ln x}{x} \stackrel{\Perp}{x}=\lim _{x \rightarrow \infty} \frac{1 / x}{1}=0
\]

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Video Transcript

okay. It's easier to rewrite access. Natural Law defeated the X for the purpose of manipulating it. When we have to apply the hoppy tolls rule you can write. This is e to the X over Max. And then we have the natural walk. Okay, Now we know we end up with the form infinity over infinity. Therefore, we know we must apply the hospital's rule. We end up with the natural log of infinity, which is simply infinity.