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

Exercise

Evaluate the following limit:

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

Final Answer

Show final answer

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

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