Math · Calculus I · Concept
Limits of functions and holes in graphs
Separate a function’s nearby behavior from its value at the point you are approaching.
Look near the point, not just at it
A limit asks what value the output approaches as the input gets close to a particular number. The function may be defined at that number, or it may not. We study nearby inputs on each side to see whether they approach the same output.
Start with the domain
The denominator is zero at x = 2, so this quotient is not defined there. Substituting 2 produces the form 0/0. That is a signal to investigate further, not a numerical answer and not a proof that the limit fails to exist.
Simplify while keeping the restriction
The difference of squares factors as (x − 2)(x + 2). For every allowed x, the factor x − 2 is nonzero and can be canceled. The simpler expression tells us what the original function does near 2; it does not fill in the missing value.
Why does cancellation help with the limit?
The two expressions agree for all inputs sufficiently close to 2 except 2 itself. A limit depends on that nearby behavior. The restriction stays attached to the original function even though x + 2, considered on its own, is defined at 2.
Check the left and right approaches
When x approaches 2 from below, x + 2 approaches 4 from below. From above, it approaches 4 from above. The two-sided limit exists because both one-sided limits agree.
| x | f(x) |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 2 | Undefined |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
Tell the graph’s story accurately
The graph follows the line y = x + 2 with a hole at (2, 4). Its limit at 2 is 4, while its value at 2 is undefined. A table or plot supports the explanation, but the algebra tells us why the nearby values behave this way.
How does this relate to continuity?
Defining a new function with value 4 at x = 2 would fill this removable hole; any other value would leave a discontinuity. The page on continuity and the Intermediate Value Theorem sets out the three conditions and the kinds of discontinuity.
Common mistakes
- Treating 0/0 as the answer instead of a reason to examine the expression.
- Canceling a factor and forgetting the excluded input.
- Using a few numerical samples as a proof for every function.
Key terms
- Limit
- The value f(x) approaches as x gets closer and closer to a point a, not counting x = a itself. A limit can exist even when f(a) is undefined or has a different value.
- Continuity
- A function is continuous at a point when it is defined there and its limit there equals that value, so the graph has no hole, jump or break. On an interval this must hold at every point, using one-sided limits at closed ends.
Work through an example
Find the limit of (x² − 4)/(x − 2) as x approaches 2.
Find a limit by factoring and canceling →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
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