Intuitive Definition of a Limit
Limit Notation
Evaluating Limits Algebraically
One-Sided Limits
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Limits of Rational Functions
Understanding the behavior of rational functions near their asymptotes is one of the most important applications of limits.
The Link to Derivatives
Limits are not just a standalone topic — they are the definition of a derivative.
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
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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