Defining Limits &
Limit Notation
Explore how functions behave as inputs approach a value, master the notation \(\displaystyle\lim_{x \to a} f(x) = L\), and build the foundation for all of calculus.
AP Calculus AB
What Is a Limit?
A limit describes the value a function \(f(x)\) approaches as the input \(x\) gets closer and closer to some number \(a\). We don't care what happens at \(x = a\): we only care about the trend nearby.
This single idea underpins derivatives, integrals, and continuity: every major concept in AP Calculus AB depends on limits.
Visualizing the Approach
Why Limits Matter
Derivatives
The derivative is defined as \(\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\).
Integrals
Definite integrals are limits of Riemann sums as the number of rectangles \(\to \infty\).
Continuity
A function is continuous at \(a\) precisely when \(\displaystyle\lim_{x\to a}f(x) = f(a)\).
Without limits, we cannot rigorously define instantaneous rates of change or areas under curves. This lesson gives you the vocabulary and notation for the rest of the course.
The Formal Notation
\(\lim\)
The limit operator: signals we are examining a trend, not computing a value.
\(x \to a\)
Read "as \(x\) approaches \(a\)." The variable \(x\) gets close to \(a\) but never equals it.
\(= L\)
The limit value: the single real number the outputs converge to.
Reading Limit Notation Aloud
1
\(\displaystyle\lim_{x\to 3} x^2 = 9\): "The limit of \(x^2\) as \(x\) approaches \(3\) equals \(9\)."
2
\(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1\): "The limit of \(\frac{\sin x}{x}\) as \(x\) approaches \(0\) equals \(1\)."
3
\(\displaystyle\lim_{x\to 5^-} f(x) = 2\): "The limit of \(f(x)\) as \(x\) approaches \(5\) from the left equals \(2\)."
One-Sided vs. Two-Sided Limits
Limit vs. Function Value
| \(\displaystyle\lim_{x\to a}f(x)\) | \(f(a)\) | |
|---|---|---|
| What it asks | Where are outputs heading? | What is the output at \(a\)? |
| Requires \(f(a)\) to exist? | No | Yes |
| Uses values near \(a\) | Yes | No |
| Can differ from each other? | Yes: the limit can exist even when \(f(a)\) is different or undefined. | |
When a Limit Does Not Exist
Different One-Sided Limits
Left and right limits approach different values: \(\lim_{x\to a^-}f(x)\neq\lim_{x\to a^+}f(x)\).
Unbounded Behavior
Outputs grow without bound: \(f(x)\to\pm\infty\) as \(x\to a\). The limit is not a finite number.
Oscillation
Outputs oscillate endlessly, like \(\sin(1/x)\) near \(x=0\), never settling on one value.
How to Evaluate a Limit
1
Direct substitution: Plug \(a\) into \(f(x)\). If you get a real number, that's the limit.
2
Indeterminate form? If you get \(\frac{0}{0}\), try factoring, rationalizing, or simplifying first.
3
Check one-sided limits: If direct algebra fails, approach from the left and right separately.
4
Confirm agreement: The two-sided limit exists only if both one-sided limits are equal.
Example 1: Direct Substitution
Find \(\displaystyle\lim_{x\to 3}\,(2x^2 - 1)\).
1
Substitute \(x = 3\): \(2(3)^2 - 1 = 2(9) - 1 = 17\).
2
No indeterminate form: the result is a finite number.
3
Conclusion: \(\displaystyle\lim_{x\to 3}(2x^2-1)=\)\(17\).
Example 2: Factoring to Resolve \(\frac{0}{0}\)
Find \(\displaystyle\lim_{x\to 2}\frac{x^2-4}{x-2}\).
1
Substitute \(x = 2\): \(\frac{4-4}{2-2}=\frac{0}{0}\). Indeterminate.
2
Factor numerator: \(x^2-4 = (x-2)(x+2)\).
3
Cancel: \(\frac{(x-2)(x+2)}{x-2}=x+2\) for \(x\neq 2\).
4
Substitute again: \(\lim_{x\to 2}(x+2)=\)\(4\).
Example 3: Reading a Graph
Example 4: One-Sided Limits from a Piecewise Function
1
Left-hand limit: Use \(f(x)=x+1\). As \(x\to 3^-\), \(f(x)\to 3+1=\)\(4\).
2
Right-hand limit: Use \(f(x)=2x-3\). As \(x\to 3^+\), \(f(x)\to 6-3=\)\(3\).
3
Compare: \(4\) \(\neq\) \(3\), so \(\displaystyle\lim_{x\to 3}f(x)\) does not exist.
Common Mistakes
✕ Plugging in and stopping at \(\frac{0}{0}\)
\(\frac{0}{0}\) is indeterminate, not "undefined." You must simplify further: factor, rationalize, or use L'Hôpital's Rule.
✕ Assuming limit = function value
The limit describes the approach, not the actual value. A function can be undefined at a point yet have a limit there.
✕ Ignoring one-sided limits
For piecewise functions or vertical asymptotes, you must check both sides. If they disagree, the two-sided limit DNE.
Limit Notation Summary
| Notation | Meaning | Reads As |
|---|---|---|
| \(\displaystyle\lim_{x\to a}f(x)=L\) | Two-sided limit equals \(L\) | "The limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\)" |
| \(\displaystyle\lim_{x\to a^-}f(x)=L\) | Left-hand limit equals \(L\) | "...as \(x\) approaches \(a\) from the left" |
| \(\displaystyle\lim_{x\to a^+}f(x)=L\) | Right-hand limit equals \(L\) | "...as \(x\) approaches \(a\) from the right" |
| \(\displaystyle\lim_{x\to a}f(x)=\infty\) | Outputs grow without bound | "The limit is infinity" (limit DNE as a finite number) |
| DNE | Limit does not exist | One-sided limits disagree, or oscillation occurs |
Question 1 of 5
What does \(\displaystyle\lim_{x\to 4}f(x)=7\) mean?
The limit notation describes the trend of \(f(x)\) as \(x\) gets close to \(4\). It says nothing about whether \(f(4)\) exists or what it equals.
Question 2 of 5
Evaluate \(\displaystyle\lim_{x\to 5}\frac{x^2-25}{x-5}\).
Factor the numerator: \(x^2-25=(x-5)(x+5)\). Cancel \((x-5)\) to get \(x+5\). Substituting \(x=5\) gives \(5+5=10\).
Question 3 of 5
If \(\displaystyle\lim_{x\to 2^-}g(x)=6\) and \(\displaystyle\lim_{x\to 2^+}g(x)=6\), what can we conclude?
When both one-sided limits exist and are equal, the two-sided limit exists and equals that common value. We cannot conclude \(g(2)=6\) or continuity without more information.
Question 4 of 5
Consider \(\displaystyle f(x)=\begin{cases}3x & x < 1 \\ 5 & x \ge 1\end{cases}\). What is \(\displaystyle\lim_{x\to 1}f(x)\)?
From the left: \(\lim_{x\to 1^-}3x = 3\). From the right: \(\lim_{x\to 1^+}5 = 5\). Since \(3 \neq 5\), the two-sided limit does not exist.
Question 5 of 5
Which of the following is not a reason a limit fails to exist?
A function can be undefined at a point and still have a limit there. The limit only cares about values near the point, not at the point itself. The other three options are genuine reasons a limit may not exist.