Question

lim x->0 (x)/(|x|)
Derivative of |x|?

Answer 1

Hypothetically, looking at the graph of \(|x|\), then the derivative would be -1 in the interval \((-\infty,0)\) and 1 in the interval \((0,\infty)\)
this is known as the signum function and is defined as:
\[\sigma(x)=\frac{|x|}{x}\]

Answer 2

Consider positive and negative values of \(x\). In other words, check the one-sided limits.

For negative \(x\), you'd be approaching 0 from the left: \(\displaystyle\lim_{x\to0^-}\frac{|x|}{x}\).

Likewise, for positive \(x\), you'd approach 0 from the right: \(\displaystyle\lim_{x\to0^+}\frac{|x|}{x}\).

What do you know about \(|x|\) for negative and positive values of \(x\) ?

Answer 3

So this limit does not exist?

Answer 4

Sorry, I meant \(\dfrac{x}{|x|}\)

Answer 5

because from the left it is -1 from the right it is 1?

Answer 6

Right, the limit doesn't exist. The derivative is given by the sign function posted earlier.

Answer 7

Thanks @SithsAndGiggles and @KeithAfasCalcLover