Evaluate lim x/|x| x-->0
To evaluate the limit of x/|x| as x approaches 0, we need to consider what happens to the function as x gets arbitrarily close to 0 from both the positive and negative sides.
To begin, let's analyze the behavior of x/|x| as x approaches 0 from the positive side. In this case, x becomes infinitesimally small positive values. The absolute value of x, |x|, remains positive.
When x is positive, the expression x/|x| simplifies to 1, as the absolute value of a positive number is equal to the number itself. So, as x approaches 0 from the positive side, x/|x| approaches 1.
Next, let's examine what happens as x approaches 0 from the negative side. In this case, x becomes infinitesimally small negative values. The absolute value of x, |x|, becomes positive since the negative sign is removed.
When x is negative, the expression x/|x| simplifies to -1, as the absolute value of a negative number is its opposite. Therefore, as x approaches 0 from the negative side, x/|x| approaches -1.
Since the limit from the positive side (approaching 0) is 1 and the limit from the negative side (approaching 0) is -1, the limit as x approaches 0 does not exist.