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Learn Limits

Understand what a function approaches, even where it is undefined.

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A limit describes the value a function gets close to as its input approaches some number, regardless of what happens exactly at that number. For example, (x squared minus 4)/(x minus 2) is undefined at x = 2, yet its values close in on 4 from both sides. Limits are the foundation of calculus: derivatives and integrals are both defined with them, and continuity is stated in terms of them. A typical unit covers reading limits from graphs and tables, algebraic techniques, one-sided limits, limits at infinity and infinite limits. Learnr builds a focused limits course and teaches each technique in short interactive lessons with instant checks.

How to learn Limits, step by step

Read limits from graphs and tables. Plug in x = 1.9, 1.99, 2.01 and 2.1 to see a function's trend near 2, and read limits off graphs with holes and jumps. This builds the idea before any algebra.
Try direct substitution first, every time. If plugging in gives a real number, that is the limit for polynomials and other continuous functions. Only when you get 0/0 do you need more work.
Learn the algebra for 0/0 forms. Factor and cancel: (x squared minus 4)/(x minus 2) becomes x + 2, so the limit at 2 is 4. For square roots, multiply by the conjugate to clear the radical from the numerator.
Handle one-sided and infinite behavior. Check 1/x as x approaches 0 from the right and from the left: positive infinity versus negative infinity, so the two-sided limit does not exist. Connect this to vertical asymptotes.
Find limits at infinity by comparing degrees. For (3x squared + 1)/(x squared minus 5), divide every term by x squared to get a limit of 3. Equal degrees give the ratio of leading coefficients; a bigger bottom degree gives 0.
Add the special limits and review daily. Learn that sin x over x approaches 1 as x goes to 0, the squeeze theorem, and L'Hopital's rule. A few limits a day in a Learnr course keeps every technique ready for derivatives.

What you'll learn

Estimate limits from graphs and tables of values
Evaluate limits by substitution, factoring and rationalizing
Find one-sided limits and limits at infinity
Decide whether a function is continuous at a point
Use L'Hopital's rule on 0/0 and infinity over infinity forms

What's inside your Limits course

A sample path Learnr builds for Limits. Your real course adapts to your goal and level.

1. What a limit means: graphs and tables
2. Limit laws and direct substitution
3. Indeterminate forms: factoring and rationalizing
4. One-sided limits, infinite limits and asymptotes
5. Limits at infinity and continuity
6. Special limits, the squeeze theorem and L'Hopital's rule

How long does it take to learn Limits?

Most learners understand limits and the standard techniques in about 2 to 4 weeks at 3 to 5 hours a week. The formal epsilon-delta definition is a separate step, usually a week or two more, and is taught in more rigorous courses.

What you need before you start

Algebra skills like factoring, simplifying fractions and rationalizing, plus familiarity with function graphs. Knowing the unit circle helps for trigonometric limits.

Common mistakes to avoid

Concluding a limit does not exist because the function is undefined at the point. A hole in the graph often has a perfectly good limit.
Writing that 0/0 equals 0 or 1. It is an indeterminate form, a signal to simplify, not an answer.
Skipping the one-sided check near vertical asymptotes and piecewise breaks, where the left and right limits can disagree.

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Limits questions, answered

Can a limit exist where the function is undefined?

Yes. A limit only depends on values near the point, not at it. (x squared minus 4)/(x minus 2) is undefined at 2 but has a limit of 4 there.

When can I use L'Hopital's rule?

Only for the indeterminate forms 0/0 or infinity over infinity. Take the derivatives of the top and bottom separately, not the quotient rule, then try the limit again.