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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


\lim _ { x \rightarrow \infty} \frac{x+\ln x}{x\ln x}=0

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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