Limit
The limit of f(x) as x approaches a is the value f(x) gets closer and closer to. The function does not need to be defined at a itself.
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Analogy
Walking towards a door: you judge where it is by your steps getting near, even before you touch it.
Example
At x = 2 this is 0/0, undefined.For x not equal 2: (x-2)(x+2)/(x-2) = x + 2.As x approaches 2, x + 2 approaches 4.So the limit is 4.
In practice
Limits let you handle 0/0 situations, which is how derivatives are built.
Practice
Flashcards
- What is lim of (x^2 - 4)/(x - 2) as x tends to 2?4. Cancel (x - 2) to get x + 2.
Interview questions
- Evaluate the limit of (x^2 - 9)/(x - 3) as x tends to 3 and explain each step.
- Substituting gives 0/0, which is indeterminate
- Factorise the top as (x - 3)(x + 3)
- Cancel the common factor for x not equal 3
- Substitute x = 3 into x + 3 to get 6
Questions people ask
Does a limit have to equal the function's value?
No. The limit describes what happens near a, not at a. The function may be undefined there, or have a different value. When the limit does equal f(a), the function is called continuous at a. Otherwise there is a gap, jump or hole in the graph.
What does it mean when a limit does not exist?
The values of f(x) near a do not settle on one number. They may jump between two values, grow without bound, or oscillate wildly. For example, 1/x near 0 grows large on both sides in opposite directions, so no single limit exists.