Evaluate the limit $\lim_{x\to\infty }\left(\frac{\sqrt{x}\ln\left(x\right)-x}{\sqrt{x}}\right)$ by replacing all occurrences of $x$ by $\infty $

Infinity to the power of any positive number is equal to infinity, so $\sqrt{\infty }=\infty$

The natural log of infinity is equal to infinity, $\lim_{x\to\infty}\ln(x)=\infty$

If you multiply a very large number by another very large number, you get an even bigger number, so infinity times infinity equals infinity: $\infty\cdot\infty=\infty$

Infinity minus infinity is an indeterminate form