Two Values to Keep Separate
A function answers a direct question: what output is assigned to a particular input? A limit asks a different question: what output values does the function approach when inputs get close to a target? These questions can have the same answer, but they do not have to.
This distinction matters whenever we interpret a graph, a table, or a formula near a particular input. A dot at the target tells us about the function’s value there. Nearby points tell us about the function’s limiting behavior. Neither piece of information automatically determines the other.
Function Value and Limiting Value
Let \(f\) be a function, and let \(a\) be a target input. The function value at the target is \(f(a)\), provided that \(a\) belongs to the domain of \(f\). If \(a\) is not in the domain, \(f(a)\) is undefined.
The limiting value at the target is the number \(L\) that the outputs \(f(x)\) approach as inputs \(x\) get arbitrarily close to \(a\), with \(x\ne a\). We write this as \(\lim_{x\to a} f(x)=L\). In this tutorial, “approach” refers to the informal limit idea introduced earlier: nearby inputs, other than the target itself, produce outputs that get close to \(L\).
The condition \(x\ne a\) is essential to the distinction. The limit concerns the function’s behavior at inputs near \(a\), not the output assigned at \(a\). Therefore, a function value can be undefined even when a limiting value exists. If \(f(a)\) is defined, it may equal the limit or differ from it.
| At the target | Nearby behavior | What can be concluded? |
|---|---|---|
| \(f(a)\) is defined | The outputs approach \(L\) | Compare \(f(a)\) with \(L\); they may agree or differ. |
| \(f(a)\) is undefined | The outputs approach \(L\) | The limit can still exist and equal \(L\). |
| \(f(a)\) is defined or undefined | The outputs do not approach one value | There is no limiting value at \(a\). |
When the Function Value and Limit Agree
Consider \(f(x)=x^2+1\) at \(a=2\). Substituting the target into the formula gives the function value. For inputs near \(2\), the outputs approach the result obtained by evaluating the same polynomial at \(2\).
Worked Example: The Function Value Matches the Limit
Let \(f(x)=x^2+1\), and compare \(f(2)\) with \(\lim_{x\to 2}f(x)\).
First evaluate at the target:
For nearby inputs, the output is \(x^2+1\). As \(x\) approaches \(2\), \(x^2\) approaches \(4\), so \(x^2+1\) approaches \(4+1=5\). Thus:
The two quantities agree: \(f(2)=5\) and \(\lim_{x\to2}f(x)=5\). As a numerical check, \(f(1.9)=1.9^2+1=3.61+1=4.61\), while \(f(2.1)=2.1^2+1=4.41+1=5.41\). These outputs lie on either side of \(5\) and move toward it as the inputs move toward \(2\). The checks illustrate the behavior; the polynomial expression establishes the limiting value.
In this example, direct substitution gives the correct limit because the formula’s nearby behavior agrees with its value at the target. It is important not to turn this observation into a rule that substitution always works. A formula may be undefined at the target, or a function may assign a target value that does not match what nearby outputs approach.
A Limit Can Exist When the Function Is Undefined
A formula can fail to give an output at the target but still have simple values at every nearby input. Factoring and simplifying can reveal this pattern. When canceling a factor, remember that the cancellation is valid only where that factor is nonzero; it does not define the original expression at the excluded input.
Worked Example: A Limiting Value Without a Function Value
For \(x\ne2\), let \(f(x)=\dfrac{x^2-4}{x-2}\). Determine the limiting value as \(x\) approaches \(2\), and decide whether \(f(2)\) exists.
At \(x=2\), the denominator is \(2-2=0\), so the original quotient does not define \(f(2)\). For nearby inputs \(x\ne2\), factor the numerator:
Every input in the domain near \(2\), other than \(2\), therefore has output \(x+2\). As \(x\) approaches \(2\), these outputs approach \(2+2=4\). Hence:
For a numerical check, at \(x=1.9\), \(f(1.9)=\dfrac{1.9^2-4}{1.9-2}=\dfrac{3.61-4}{-0.1}=\dfrac{-0.39}{-0.1}=3.9\). At \(x=2.1\), \(f(2.1)=\dfrac{2.1^2-4}{2.1-2}=\dfrac{4.41-4}{0.1}=\dfrac{0.41}{0.1}=4.1\). These outputs are close to \(4\). The cancellation describes nearby inputs; it does not remove the zero denominator in the original formula at \(x=2\).
A Defined Function Value Can Differ from the Limit
A function may be defined at the target and still assign a value different from the value approached nearby. In the next example, the formula for inputs other than the target simplifies to a simple expression, while the function’s stated value at the target is different. The comparison is between that assigned value and the outputs near the target.
Worked Example: The Function Value Differs from the Limit
Define \(f\) by
First read the value assigned at the target directly from the definition:
To find the limiting value, use the formula for nearby inputs \(x\ne1\). Factor and cancel only the nonzero factor \(x-1\):
As \(x\) approaches \(1\), the nearby outputs \(x+1\) approach \(1+1=2\). Therefore:
For a check, \(f(0.9)=\dfrac{0.9^2-1}{0.9-1}=\dfrac{0.81-1}{-0.1}=\dfrac{-0.19}{-0.1}=1.9\). Also, \(f(1.1)=\dfrac{1.1^2-1}{1.1-1}=\dfrac{1.21-1}{0.1}=\dfrac{0.21}{0.1}=2.1\). The nearby outputs are close to \(2\), whereas the function value at \(1\) is \(10\). Thus the limit and function value differ.
Why the Target Value Does Not Determine the Limit
The reasoning in the examples rests on a basic principle: to determine a limit, inspect what the function does at inputs near the target, excluding the target itself. The value at the target is not part of that nearby-input calculation. This is why knowing \(f(a)\) alone is not enough to determine \(\lim_{x\to a}f(x)\).
For the quotient in the previous example, the expression \(x+1\) describes every output for \(x\ne1\). Those are exactly the inputs relevant to the limiting behavior. The assigned output \(f(1)=10\) does not change any of those nearby outputs. Consequently, the limit remains \(2\), not \(10\).
Likewise, in the rational example that was undefined at \(2\), the factorization showed that every allowed nearby input produces the output \(x+2\). The absence of an output at \(2\) does not alter those nearby values. This gives a limiting value of \(4\) even though the function value at the target does not exist.
A Common Mistake: Substituting Without Checking
A frequent error is to treat direct substitution as a universal method for finding limits. For example, in the quotient \(\dfrac{x^2-4}{x-2}\), substituting \(x=2\) gives:
The incorrect conclusion is “the limit is \(0/0\), so there is no limit.” But \(0/0\) is not a number, and the calculation has only shown that the original formula does not give a value at \(x=2\). It has not determined what the outputs do nearby.
The correction is to examine the expression at allowed nearby inputs:
Since \(x+2\) approaches \(4\) as \(x\) approaches \(2\), the limit is \(4\). The original function value at \(2\) remains undefined. A failed substitution can be a reason to examine the formula more carefully, not a final answer about the limit.
A Reliable Comparison Process
When a problem asks you to compare a function value and a limiting value, keep the two questions separate. This prevents a target value from being mistaken for nearby behavior, or an undefined formula value from being mistaken for a failed limit.
Use the function definition at \(a\). State \(f(a)\) if defined; otherwise state that it is undefined.
Use the formula valid for \(x\ne a\), simplifying only when the algebraic steps are valid there.
Determine the value that the nearby outputs approach as \(x\) approaches \(a\).
Say whether \(f(a)\) equals the limit, differs from it, or is undefined.
This process is useful because the two quantities answer different questions. It also forces you to state exactly what is known: a value at one input, a pattern in nearby outputs, or both.
Check Your Understanding
For each question, keep the function value and limiting value separate. Explain how you know your conclusion.
- If \(g(x)=3x-1\), what are \(g(2)\) and \(\lim_{x\to2}g(x)\)? Do they agree?
- For \(h(x)=\dfrac{x^2-9}{x-3}\), defined where \(x\ne3\), is \(h(3)\) defined? What limiting value do nearby outputs approach?
- A function has \(p(4)=7\), while its outputs approach \(2\) as \(x\) approaches \(4\). Which is the function value, and which is the limiting value?
- Why does obtaining \(0/0\) by substitution into a quotient not prove that its limit fails to exist?
- Can a limiting value exist when the function is undefined at the target? Give a reason based on nearby inputs.
