# Calculating Limit of Function- A function with ln in the power of x to 0 from right – Exercise 6323

### Exercise

Evaluate the following limit:

$$\lim _ { x \rightarrow 0^+} {(\ln \frac{1}{x})}^x$$

$$\lim _ { x \rightarrow 0^+} {(\ln \frac{1}{x})}^x=0$$

### Solution

First, we try to plug in $$x = 0^+$$ and get

$${(\ln \frac{1}{0^+})}^{0^+}$$

We got the phrase $$\infty^0$$ (=infinity in the power of zero). This is an indeterminate form, therefore we have to get out of this situation.

$$\lim _ { x \rightarrow 0^+} {(\ln \frac{1}{x})}^x=$$

Using Logarithm Rules we get

$$=\lim _ { x \rightarrow 0^+} e^{\ln{(\ln \frac{1}{x})}^x}=$$

$$=\lim _ { x \rightarrow 0^+} e^{x\ln(\ln \frac{1}{x})}=$$

We enter the limit inside:

$$= e^{\lim _ { x \rightarrow 0^+} x\ln(\ln \frac{1}{x})}=$$

Note: This can be done because an exponential function is a continuous function.

We plug in 0+ again and get

$$= e^{ 0^+\cdot\ln(\ln \frac{1}{0^+})}=$$

$$= e^{ 0^+\cdot\ln(\ln \infty)}=$$

$$= e^{ 0^+\cdot\infty}$$

We got the phrase $$0\cdot\infty$$(=tending to zero multiples by infinity). This is also an indeterminate form, it such cases we use Lopital Rule – we derive the numerator and denominator separately and we will get

In order to use Lopital Rule, we simplify the phrase and get:

$$= e^{\lim _ { x \rightarrow 0^+} \frac{\ln(\ln \frac{1}{x})}{\frac{1}{x}}}=$$

We plug in 0+ again and get

$$= e^{\frac{\ln(\ln \frac{1}{0^+})}{\frac{1}{0^+}}}=$$

$$= e^{\frac{\infty}{\infty}}=$$

We got the phrase $$\frac{\infty}{\infty}$$ (=infinity divides by infinity). This is also an indeterminate form, in such cases we use Lopital Rule – we derive the numerator and denominator separately and we will get

$$= e^{\lim _ { x \rightarrow 0^+} \frac{\frac{1}{\ln\frac{1}{x}}\cdot \frac{1}{\frac{1}{x}}\cdot (-\frac{1}{x^2})}{-\frac{1}{x^2}}}=$$

We simplify the phrase and get

$$= e^{\lim _ { x \rightarrow 0^+} \frac{1}{\frac{1}{x}\ln\frac{1}{x}}}=$$

We plug in 0+ again and this time we get

$$= \frac{1}{\frac{1}{0^+}\ln\frac{1}{0^+}}=$$

$$= \frac{1}{\infty\cdot\ln\infty}=$$

$$= \frac{1}{\infty\cdot\infty}=$$

$$= \frac{1}{\infty}=$$

$$= 0$$

Note: A finite number divides by infinity is defined and equals to infinity. For the full list press here

Have a question? Found a mistake? – Write a comment below!
Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions!

Share with Friends