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

### Exercise

Evaluate the following limit:

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

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

### 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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