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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


\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

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