Question

If f(x)={ ∣x∣+1, 0, x<0}

{ x3 x=0}
{ ∣x∣−1 x>0}
​For what value (s) of a does lim x→a f(x) exists?

Answer 1

The limit of the piecewise function f(x) exists for all a ≠ 0, meaning for all real numbers except zero because at x = 0 the left-hand limit and the right-hand limit are not equal.

Step-by-step explanation:

The student asks for which value(s) the limit of the piecewise function f(x) exists as x approaches a.

To determine the existence of the limit, one must consider the behavior of f(x) as it approaches from the left and right around the point in question. Since f(x) is described differently for values of x less than zero, equal to zero, and greater than zero, we have to examine the limits separately in these ranges.

For all a less than zero, f(x) = |x| + 1, which is a continuous function for x less than zero, and thus the limit exists.

At x = 0, we need to see if the right-hand limit and the left-hand limit both equal f(0), which is equal to zero. The left-hand limit as x approaches 0 is |x| + 1 = 1, and the right-hand limit is |x| - 1 = -1 (given x approaches 0 from the positive side), hence, the limit at x = 0 does not exist since the left-hand limit and the right-hand limit are not equal.

For all a greater than zero, f(x) = |x| - 1, a continuous function for x greater than zero, ensuring the limit exists.

Therefore, the limit of f(x) exists for all a ≠ 0, meaning all real numbers except zero.