Evaluate lim x/|x| x-->0
To evaluate the limit lim(x→0) x/|x|, we first need to determine the behavior of the function |x| as x approaches 0. The function |x| represents the absolute value of x, which is defined as x when x is greater than or equal to 0, and -x when x is less than 0.
Considering this definition, as x approaches 0 from the positive side (x > 0), |x| is equal to x. Thus, in this case, the expression x/|x| simplifies to x/x, which equals 1.
On the other hand, as x approaches 0 from the negative side (x < 0), |x| is equal to -x. In this situation, the expression x/|x| simplifies to x/(-x), which simplifies to -1.
Since the limit is evaluated from both sides and produces different results, lim(x→0) x/|x| does not exist.