A Limit Is Where Nearby Values Are Heading
To understand a two-sided limit, watch what happens on both sides of the target. The points do not need to land on the target; they only need to get as close as we want while their outputs settle toward the same number.
Move the idea
Try it: bring both points toward the target
Change one parameter and watch what changes with it.
A two-sided limit exists when left-side and right-side behavior agree on the same destination.
Approach from the left
Inputs such as 0.9, 0.99 and 0.999 produce outputs closer and closer to the target value.
Approach from the right
Inputs such as 1.1, 1.01 and 1.001 must head toward the same output. If the two sides disagree, the two-sided limit fails.
Substitution comes later
Direct substitution is convenient when the function is continuous, but the definition of a limit is about nearby behavior, not the substitution shortcut.