Mathematics

Problem

The graph of a function

g

has an open circle at

(1,4)

and another open circle at

(1,6)

. As

x

approaches

1

from the left,

g(x)

approaches

4

, and as

x

approaches

1

from the right,

g(x)

approaches

6

. Explain whether

\displaystyle \lim_{x\to 1} g(x)

exists and give its value if it does.

Intuitive explanation

A limit at a point exists only when the function approaches the same value from both sides—think of it as a one-way street where both directions must lead to the same destination. Here, the left-hand limit is $4$ and the right-hand limit is $6$ , which are different, so the two-sided limit cannot exist. The open circles tell you the function isn't actually defined at $x=1$ , but that doesn't matter for limits; what matters is the behavior near the point, and that behavior is inconsistent.

Intuitive explanation

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