How do left-hand and right-hand limits reveal a jump?

You are comparing the y-values the function gets close to as x gets close to a chosen point. For the left-hand limit, approach the point using x-values smaller than it. For the right-hand limit, use x-values larger than it. [S1]

Keep the direction separate from the function’s value at the point. A limit asks what the nearby outputs approach; it does not simply ask what the function equals when x is exactly the chosen number. A useful first step is to write the point and the two directions down, so you don’t accidentally use the same piece of a formula for both sides.

Suppose, for this example, the function is

f(x) = x + 1 when x < 2 f(x) = 5 when x ≥ 2

We want to compare the limits as x approaches 2.

1. From the left, x-values are less than 2, so use f(x) = x + 1. As x gets close to 2, x + 1 gets close to 3. Thus the left-hand limit is 3.
2. From the right, x-values are greater than 2, so use f(x) = 5. The outputs stay at 5, so the right-hand limit is 5.
3. Compare: 3 ≠ 5. Because the one-sided limits disagree, the two-sided limit at 2 does not exist. [S1]

Here the two sides approach different finite values, so the graph has a jump-type break at x = 2. The example’s function value is f(2) = 5, because the second rule includes x = 2. That value matches the right-hand limit, but it does not make the left-hand limit equal 5. Different one-sided limits show that the two-sided limit does not exist; the finite, differing limits in this example give a jump-type break.

Read the conditions attached to each piece before substituting anything. In the example, “x < 2” governs values to the left, while “x ≥ 2” governs the value at 2 and values to the right. For the right-hand limit, x is greater than 2, so the second rule applies. For the left-hand limit, the first rule applies.

You can also check with nearby test inputs, without treating them as the proof: 1.9 uses x + 1, while 2.1 uses 5. Those outputs suggest the two approaches. The limit calculation comes from the formulas and their behavior as x gets close to 2, not just from one nearby value.

State both one-sided limits, compare them, and then give the conclusion. For the worked example, you could write:

“Using the rule for x < 2, lim as x → 2⁻ of f(x) = 3. Using the rule for x > 2, lim as x → 2⁺ of f(x) = 5. Since 3 ≠ 5, lim as x → 2 of f(x) does not exist. The function approaches different finite values from the two sides, giving a jump-type break.”

The left arrow marker 2⁻ means approach from below; 2⁺ means approach from above. Keep those results visible in your work so the reader can see why the two-sided limit fails.