Question

Just need a little explanation please

Answer 1

\[\lim_{x \rightarrow 0^{+}} x ^{x} \]
i Know how to solve these, but i never understood why does the e goes under lim, even though it is a constant
\[e ^{\lim_{x \rightarrow ^{+}}} xlnx \]

Answer 2

Recall that the exponential is the inverse of the log.

Whenever you take the composition of inverses,
`they undo one another`.

This is very evident when you look at simple operations like addition or multiplication.

If I take an expression like \(\large\rm 5x\) then applying the inverse of multiplication, division, will undo the multiplication, right? \(\large\rm \frac{5x}{5}=x\).

Same with trig stuff, \(\large\rm \sin(arcsin(x))=x\)
Taking the sine of the inverse sine of a value x, they undo one another.

Same with our exponential and log.\[\large\rm e^{\ln x}=x\]They undo one another.

We want to apply this idea in reverse.

Answer 3

\[\large\rm \color{orangered}{x}=e^{\ln \color{orangered}{x}}\]So for our problem,\[\large\rm \color{orangered}{\lim_{x\to0^+}x^x}=e^{\ln\left(\color{orangered}{\lim_{x\to0^+}x^x}\right)}\]

Answer 4

And then we pass the limit notation out of the log,\[\large\rm =e^{\lim_{x\to0^+}\ln(x^x)}\]And then apply some cool tricks from there.

Answer 5

Thank u so much for the explanation