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Free calculus explanations

When does a two-sided limit not exist?

Imagine two carts moving toward the same x-position. If they arrive at different heights, there is no single height for the two-sided limit.

Quick answer: A two-sided limit does not exist when the values approaching from the left and right do not settle on the same number. A jump is a clear example.

A jump at x = 2

Consider a function with f(x) = x + 1 when x < 2 and f(x) = x + 3 when x > 2. We do not need to define f(2) to find either one-sided limit.

From the left, x + 1 heads toward 3. From the right, x + 3 heads toward 5.

limx→2⁻ f(x) = 3   and   limx→2⁺ f(x) = 5.

Because 3 ≠ 5, the two-sided limit at x = 2 does not exist. Assigning a value to f(2) would not make the two sides agree.

At x equals 2, the left branch approaches y equals 3 and the right branch approaches y equals 5left → 3right → 5x = 2

A quick test for any graph

  1. Follow the graph toward the target x-value from smaller x-values.
  2. Follow it again from larger x-values.
  3. Compare the heights. If they match, that shared number is the two-sided limit. If they differ, the limit does not exist.

A hole by itself does not make a limit fail. Compare this jump with a removable hole whose limit is 4.

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