Exercise
Evaluate the following limit:
\lim _ { x \rightarrow \infty} \frac{x+\ln x}{x\ln x}
Final Answer
Solution
First, we try to plug in x = \infty and get
\frac{\infty+\ln \infty}{\infty\ln \infty}=\frac{\infty}{\infty}
In the base we got the phrase \frac{\infty}{\infty} (=infinity divides by infinity). This is an indeterminate form, therefore we have to get out of this situation.
\lim _ { x \rightarrow \infty} \frac{x+\ln x}{x\ln x}=
In such cases we use Lopital Rule – we derive the numerator and denominator separately and we will get
=\lim _ { x \rightarrow \infty} \frac{1+\frac{1}{x}}{\ln x+1}=
We plug in infinity again and get
= \frac{1+\frac{1}{\infty}}{\ln \infty+1}=
\frac{1}{\infty}=
=0
Note: Any finite number divides by infinity is defined and equals to zero. For the full list press here
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