Exercise
Evaluate the following limit:
\lim _ { x \rightarrow 0^+}x^n\ln x
Final Answer
Solution
First, we try to plug in x = 0^+
When we aim for zero on the right, we are close to zero, but are larger than zero (e.g., 0.00001), therefore we get
{(0^+)}^n\ln (0^+)
We got the phrase "0"\cdot (-\infty) (=tends to zero multiplies by minus infinity). This is an indeterminate form, therefore we have to get out of this situation.
Note: The zero in the expression is not absolute zero, but a number that is tending to zero. That is why the expression is not equal to zero.
In order to use Lopital Rule, we simplify the expression and get
\lim _ { x \rightarrow 0^+}x^n\ln x=
=\lim _ { x \rightarrow 0^+}\frac{\ln x}{x^{-n}}
Now, if we plug in zero in the phrase. we will get \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 get
=\lim _ { x \rightarrow 0^+}\frac{\frac{1}{x}}{-nx^{-n-1}}=
We simplify the expression
=\lim _ { x \rightarrow 0^+}\frac{1}{\frac{-n}{x^n}}=
We plug in zero again and this time we get
=\frac{1}{\frac{-n}{{(0^+)}^n}}=
=\frac{1}{\frac{-n}{0^+}}=
=\frac{1}{-\infty}=
=0
Note: The zero in the denominator is not an absolute zero, but a number tending to zero. A finite number divides by a number tending to zero is defined and equals to infinity. For the full list press here
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