MHB Solving $\lim_{{x}\to{0}} lnx \cdot x$: Overview & Proofs

Thread starter tmt1
Start date Dec 4, 2016
Tags

Limit

AI Thread Summary

The limit $\lim_{{x}\to{0}} \ln x \cdot x$ involves $x$ approaching 0 while $\ln x$ approaches $-\infty$. The reasoning suggests that $x$ approaches 0 faster than $\ln x$ approaches $-\infty$, leading to the conclusion that the limit is 0. To prove this, the expression can be rewritten as $L=\lim_{x\to0}\left(\frac{\ln(x)}{\frac{1}{x}}\right)$, which allows for the application of L'Hôpital's Rule. This method confirms that the limit indeed evaluates to 0. The final conclusion is that the limit is 0.

Dec 4, 2016

tmt1

I have

$\lim_{{x}\to{0}} lnx \cdot x$

$x$ approaches 0, and $lnx$ approaches $\infty$.

How can I reason about this.

I suppose $x$ approaches 0 more quickly than $lnx$ approaches $\infty$ , therefore it is zero. Is this accurate? How can I prove this.

Mathematics news on Phys.org

Dec 4, 2016

MarkFL

Gold Member

MHB

I would write it as:

$$L=\lim_{x\to0}\left(\frac{\ln(x)}{\dfrac{1}{x}}\right)$$

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

View full post »

Thread 'Midpoint(s) of the unbounded number line'

Just chatting with my son about Maths and he casually mentioned that 0 would be the midpoint of the number line from -inf to +inf. I wondered whether it wouldn’t be more accurate to say there is no single midpoint. Couldn’t you make an argument that any real number is exactly halfway between -inf and +inf?

View full post »

Thread 'Arithmetic and our place value system'

A power has two parts. Base and Exponent. A number 423 in base 10 can be written in other bases as well: 1. 4* 10^2 + 2*10^1 + 3*10^0 = 423 2. 1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 = 1143 3. 7*60^1 + 3*60^0 = 73 All three expressions are equal in quantity. But I have written the multiplier of powers to form numbers in different bases. Is this what place value system is in essence ?

View full post »