Question

(Continuity) What is the limit of abs(x)/x as x approaches 0 from the right?

Answer 1

What is abs(x)/x when x is a very small positive number?

Answer 2

Since x approaches from the right
IxI=x

Answer 3

1?

Answer 4

Yep

Answer 5

thank you!

Answer 6

Welcome

Answer 7

The obvious next question is going to be: What is the limit of abs(x)/x as x approaches 0 from the left? To this the same thinking goes into: "What is abs(x)/x when x is a very small (in magnitude) negative number?"

The general point is that the limit from the left will be different from the limit from the right. Hence the function abs(x)/x is discontinuous at x=0. (Continuity requires those limits to equal each other).

Answer 8

\[\ |x| = \left\{\begin{array}{rcc}
-x & \text{if} & x <0 \\
x& \text{if} &\geq 0
\end{array}
\right.
\]
and so
\[\frac{|x|}{x} = \left\{\begin{array}{rcc}
-1 & \text{if} & x<0 \\
1& \text{if} & x >0
\end{array}
\right.
\]

Answer 9

I understand. Thank you