Intuitive Definition of a Limit
A limit describes what value a function approaches as its input gets arbitrarily close to some target. It does NOT describe what the function equals at the target — those can be different or the function may not be defined there at all.
Imagine walking toward a door. Your distance from the door gets smaller and smaller. The limit of your distance as you approach the door is zero — even if you never actually touch the door. Limits formalize this idea of "approaching without necessarily reaching."
Example: \lim_{x \to 2} (3x+1) = 7. As x gets close to 2, the value of 3x + 1 gets close to 7. Simple — but the power of limits appears when the function is not defined at the target point, yet still approaches a value.
Limit Notation
Standard limit notation:
Read as: "the limit of f(x) as x approaches a equals L." The arrow → means "approaches"; a can be a finite number, positive infinity, or negative infinity.
The limit L must be a specific real number for the limit to "exist." When the function blows up to infinity or oscillates indefinitely, the limit does not exist (DNE).
Evaluating Limits Algebraically
Step 1 — Direct substitution. For continuous functions (polynomials, sine, cosine, exponentials, etc.), just plug in the target value:
Step 2 — Factor and cancel. If substitution gives the indeterminate form 0/0, try factoring. Cancel the common factor, then substitute.
Example: Evaluate \lim_{x \to 2} \dfrac{x^2-4}{x-2}. Direct substitution gives 0/0, so factor:
Step 3 — Rationalize. For limits involving square roots, multiply by the conjugate to eliminate the radical:
Example: Evaluate \lim_{x \to 0} \dfrac{\sqrt{x+1} - 1}{x}. Multiply top and bottom by √(x+1) + 1:
One-Sided Limits
A one-sided limit considers only one direction of approach: x → a⁻ (from the left, smaller values) or x → a⁺ (from the right, larger values). The two-sided limit exists only when both one-sided limits agree.
Example: For the piecewise function f(x) = \begin{cases} x+1 & x < 2 \\ x^2 & x \geq 2 \end{cases}:
- \lim_{x \to 2^-} f(x) = 3 (uses x + 1)
- \lim_{x \to 2^+} f(x) = 4 (uses x²)
Since 3 ≠ 4, the two-sided limit at x = 2 does not exist.
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Try an interaction checkInfinite Limits and Limits at Infinity
Infinite limits describe functions that grow without bound as x approaches a finite value — this is how limits detect vertical asymptotes:
Limits at infinity describe end behavior — what happens as x grows without bound. These detect horizontal asymptotes:
Shortcut for rational functions at infinity: compare numerator and denominator degrees. Equal degrees give the ratio of leading coefficients. Smaller numerator degree gives 0. Larger numerator degree gives ±∞.
Limits of Rational Functions
Understanding the behavior of rational functions near their asymptotes is one of the most important applications of limits.
At a vertical asymptote: the one-sided limits blow up to +∞ or -∞. Check signs from each side to determine which.
At a hole (removable discontinuity): the limit exists and equals the value of the simplified function, even though the original function is undefined there. This is exactly when factor-and-cancel works.
At infinity: the limit equals the horizontal asymptote (or ±∞ if there is an oblique asymptote or higher-degree growth).
The Link to Derivatives
Limits are not just a standalone topic — they are the definition of a derivative:
Every derivative rule (power rule, product rule, chain rule) is proved using this limit definition. The same 0/0 indeterminate form you saw in factor-and-cancel examples is exactly what you get when computing a derivative directly from the definition — which is why the algebraic techniques of this section are so important.
Common Misunderstandings
Thinking the function must equal the limit
A limit describes what the function approaches, not what it equals. The function can have a different value (or be undefined) at the limit point itself.
Assuming 0/0 means the limit is 0 or undefined
0/0 is an indeterminate form — it tells you that more work is needed (factoring, L'Hôpital's rule, etc.), not that the limit is zero or does not exist.
Practice Problems
Evaluate each limit. Try direct substitution first; use factoring or rationalizing when you get 0/0.
- \lim_{x \to 4} (x^2 - 3x + 1)
- \lim_{x \to 3} \dfrac{x^2-9}{x-3}
- \lim_{x \to 1} \dfrac{x^2-1}{x^2-x}
- \lim_{x \to 0} \dfrac{\sqrt{x+4}-2}{x}
- \lim_{x \to \infty} \dfrac{2x^3 - x + 1}{5x^3 + 7}
- \lim_{x \to \infty} \dfrac{x+1}{x^2-1}
- \lim_{x \to 2^+} \dfrac{1}{x-2}
Answers
- 5 (direct substitution)
- 6 (factor and cancel)
- 2 (factor: (x-1)(x+1) / (x(x-1)) → (x+1)/x → 2)
- 1/4 (rationalize)
- 2/5 (equal degrees: ratio of leading coefficients)
- 0 (denominator degree larger)
- +∞ (vertical asymptote, right-hand blow-up)
Related Guides
Frequently Asked Questions
What is a limit in calculus?
A limit describes what value a function approaches as the input approaches a certain point. It does not require the function to actually reach that value — it only asks what the output gets closer and closer to. Limits are the foundation of both derivatives and integrals.
What is the difference between a limit existing and a function being defined?
A function can have a limit at a point where it is not defined. For example, (x²-1)/(x-1) is not defined at x=1, but its limit as x approaches 1 is 2 (because the expression simplifies to x+1). Conversely, a function can be defined at a point where the limit does not exist.
What are one-sided limits?
One-sided limits examine what happens when you approach a point from only one direction. The left-hand limit (x → a⁻) approaches from values less than a; the right-hand limit (x → a⁺) approaches from values greater than a. The two-sided limit exists only if both one-sided limits exist and are equal.
What is an infinite limit?
An infinite limit occurs when the function grows without bound as x approaches a certain value. We write lim f(x) = ∞ or -∞. This typically happens at vertical asymptotes of rational functions. Technically, the limit "does not exist" in the finite sense, but the infinity notation describes the specific behavior.
How are limits related to derivatives?
A derivative is defined as a limit: f'(x) = lim[h→0] (f(x+h) - f(x))/h. This limit gives the instantaneous rate of change — the slope of the tangent line. Without limits, derivatives cannot be rigorously defined. Understanding limits is prerequisite to understanding derivatives.
What does "the limit does not exist" mean?
A limit does not exist when the function does not approach a single finite value. This can happen when: the left and right limits are different, the function oscillates infinitely, or the function approaches infinity. Each case has a different interpretation.
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