A limit describes what a function approaches as the input gets close to a value. Sometimes direct substitution gives the answer immediately. Other times it gives an indeterminate form, which means more work is needed.
Try substitution first
For many limits, substitute the target value. If the expression becomes a normal number, that is the limit. If it becomes \(0/0\), infinity over infinity, or another indeterminate form, simplify before deciding.
Common indeterminate forms
The most familiar form is \(0/0\). It does not mean the answer is zero or undefined. It means the expression has competing behavior and must be simplified. Other forms include infinity over infinity, zero times infinity, and infinity minus infinity.
Factoring example
Evaluate \(\lim_{x\to2} \frac{x^2 - 4}{x - 2}\). Direct substitution gives \(0/0\). Factor the numerator as \((x - 2)(x + 2)\), cancel \(x - 2\) for \(x \ne 2\), and evaluate \(x + 2\) at \(x = 2\). The limit is \(4\).
Other strategies
Rationalizing helps with square roots. Known special limits help with trigonometric expressions. Dividing by the highest power of \(x\) helps with rational functions as \(x\) approaches infinity.
Common mistakes
Do not cancel terms that are not factors. Do not stop at \(0/0\). Do not assume the function value at the point must equal the limit.
Choosing a simplification method
Once you see an indeterminate form, choose the simplification that matches the expression. Polynomials often factor. Square roots often need rationalizing. Trigonometric limits may need known standard limits. Rational functions at infinity often simplify by dividing by the highest power of the variable.
This topic connects directly to One-Sided Limits and Discontinuities. If the expression has a denominator that becomes zero, check whether the left and right sides behave the same way. If they do not, a two-sided limit may not exist even if one side has a clear behavior. Limit work also supports derivatives, because the derivative is built from a limit definition; for derivative practice, see Derivative Rules Every Student Should Know.
When writing a solution, do not simply state that the answer is found by canceling. Show the factorization or transformation that made cancellation legal. This is important because canceling terms that are not common factors is one of the most frequent limit errors.
Calculator support
Use the Limit Calculator to compare simplification steps, and use the one-sided limit calculator when behavior differs from the left and right.
Study checklist
For limits, write down what direct substitution gives before choosing a method. A normal number ends the problem. An indeterminate form starts the real work. This habit makes your solution easier to follow and shows why factoring, rationalizing, or another transformation is necessary.
When an answer does not exist, explain why. Does the expression grow without bound? Do the one-sided limits disagree? Is the function oscillating? Different reasons lead to different mathematical statements, and writing the reason makes the answer stronger than simply saying DNE.
Practice prompts
Make a table of limits and the first result from substitution. Use columns for the expression, substitution result, method, and final limit. Some rows should produce ordinary numbers, while others should produce \(0/0\) or unbounded behavior. This helps you see that not every limit requires the same amount of work.
For each indeterminate example, write why the chosen method is valid. Factoring is valid when a common factor appears. Rationalizing is useful when square roots create cancellation. Dividing by the highest power works for rational expressions at infinity. Naming the method makes the solution easier to review later.
Teacher-style review questions
What does direct substitution give? If the result is indeterminate, what algebraic feature suggests the next move? Are there hidden domain restrictions near the approach point? Does the left side behave like the right side? These questions make limit work more precise and reduce the habit of treating every \(0/0\) problem the same way.
Quick self-test
Before moving on, write one original example and one sentence explaining the method. Then change a small part of the example and predict whether the same method still works. This tiny variation exercise is a strong test of understanding because it separates memorized answers from flexible problem solving.
For deeper review, write a short explanation as if teaching another student. Include what the problem is asking, which clue tells you the method, what the first algebraic step should be, and what kind of final answer is expected. This turns the guide into active practice and helps the page serve real learners with reasoning, examples, and study support instead of only final answers.