Calculating Limit of Function – A quotient of functions with ln to infinity – Exercise 6305

Exercise

Evaluate the following limit:

$\lim _ { x \rightarrow \infty} \frac{x+\ln x}{x\ln x}$

Final Answer

Show 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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