A limit is where a path is heading, not where it is. Pip walks toward a hole in the path, gets as close as he likes, and never steps in. Slide the distance and watch the values close in on the limit.
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To estimate a limit, evaluate the function at inputs closer and closer to the point. For (x² − 4) ÷ (x − 2), the values at x = 1.9, 1.99 and 1.999 are 3.9, 3.99 and 3.999.
They head for 4, from both sides. Stopping at one value is the usual mistake: the limit is what the whole trail is heading for.
A limit is where the path is heading, not where it is.
The limit only cares where the path is heading. The value at the point does not matter: it can be missing, as with a hole, or sit somewhere else, as with a stray dot.
Here the path heads for 4 at x = 2, but the function’s value at 2 is 1. The limit is still 4. Giving f(2) instead of the limit is the usual slip.
The value at the point does not matter: the limit is where the path is heading.
Approach a point from the left and from the right. If both sides head for the same number, that number is the limit.
If they head for different numbers, the path jumps and the limit does not exist at that point. Saying the limit exists when the sides differ is the usual mistake.
If the two sides head for different places, there is no limit.
Far away, the path 1 ÷ x settles toward 0: a limit at infinity, and a horizontal asymptote. Near x = 0 it shoots off with no ceiling: a vertical asymptote.
From the right, 1 ÷ x grows without bound. From the left it falls without bound. A curve can settle toward a level and, unlike a wall, can even cross it on the way.
Far away the path settles to a level, and near a wall it shoots off with no ceiling.
The value a function heads toward as the input gets closer to a point. It describes where the path is heading, not necessarily where it is.
Yes. For (x² − 1) ÷ (x − 1) at x = 1 the function is undefined, but the limit is 2, because the path heads for 2 from both sides.
It is an indeterminate form. It does not mean the limit is 0, and it does not mean the limit does not exist. Simplify first, for example by cancelling a common factor, then put the number in.
When the path is unbroken: no hole, no jump and no wall. The function is defined at the point, the limit exists there, and the two are equal.
World 9 of Pip: Math Adventure is about limits. You close the gap in Pip’s path, and learn to find where a function is heading near a point and far away.
It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.