Limits describe the value a function approaches as the input gets close to a number, even when the function is not defined there. This cheat sheet helps students organize the main rules for evaluating limits quickly and accurately. It also connects limits to continuity, which is one of the foundations of calculus.
Students need these ideas before learning derivatives, tangent lines, and many real-world rate problems.
Core concepts include direct substitution, algebraic simplification, one-sided limits, infinite limits, and special limit theorems. A function is continuous at when exists, exists, and . Important formulas include the quotient law when the denominator limit is not .
The Intermediate Value Theorem explains why continuous functions take every value between two endpoint values on a closed interval.
Key Facts
- Direct substitution works when is continuous at , so .
- The sum and difference laws state that when both limits exist.
- The product law states that when both limits exist.
- The quotient law states that when .
- A two-sided limit exists only if .
- A vertical asymptote occurs at if or .
- The Squeeze Theorem says if near and , then .
- Continuity at requires to exist, to exist, and .
Vocabulary
- Limit
- A limit is the value that approaches as gets close to a number .
- One-sided limit
- A one-sided limit describes what approaches as approaches from only the left or only the right.
- Continuity
- Continuity at means the graph has no hole, jump, or break there, and .
- Removable discontinuity
- A removable discontinuity is a hole in the graph where the limit exists but the function value is missing or different.
- Jump discontinuity
- A jump discontinuity occurs when and both exist but are not equal.
- Vertical asymptote
- A vertical asymptote is a line where the function grows without bound toward or .
Common Mistakes to Avoid
- Substituting immediately into every limit is wrong because expressions such as are indeterminate and need simplification first.
- Assuming a hole changes the limit is wrong because a limit depends on nearby values of , not only on the value of .
- Combining one-sided limits without checking equality is wrong because exists only when .
- Treating as is wrong because division by is undefined, and the expression may indicate an infinite limit or no limit.
- Calling a function continuous just because it is defined at is wrong because continuity also requires to exist and equal .
Practice Questions
- 1 Evaluate .
- 2 Evaluate .
- 3 For , determine whether exists and whether is continuous at .
- 4 Explain why a function can have a limit at even if it is not continuous at .
Understanding Limits & Continuity
A limit is about nearby behavior, so the point itself can be misleading. A graph may have an open circle at one height and a filled dot at another height. The open circle shows the value suggested by nearby inputs.
The filled dot shows the function's assigned value at that exact input. These can differ without changing the limit. This distinction matters when a formula has been simplified.
For example, factoring may cancel a factor that causes a hole, but the original formula still excludes that input. Keep the original domain restrictions even after simplifying an expression.
When substitution produces zero divided by zero, it does not mean the answer is zero. It signals that the expression needs more work. Factoring is useful for polynomials.
Rationalizing is useful when square roots are involved. A common denominator can help with complex fractions. After simplifying, evaluate the behavior of the new expression near the target input, while remembering any excluded values from the original form.
By contrast, a nonzero number divided by zero often indicates values growing without bound. The sign of nearby factors tells whether the graph rises toward positive infinity or falls toward negative infinity on each side.
One-sided thinking is especially important for piecewise graphs and real situations with boundaries. A parking garage fee might follow one rule before a time cutoff and another rule after it. A two-sided limit cannot be decided from only one direction.
Make a small table using inputs slightly smaller and slightly larger than the target, or trace the graph from both sides. Tables can suggest an answer, but they do not prove one by themselves because rounding can hide important behavior.
Exact algebra and careful graph reading give stronger evidence. Watch for jumps, holes, and vertical asymptotes, since each represents a different kind of break.
Continuity matters because it supports the idea of change without sudden gaps. A continuous model is often reasonable for height, distance, temperature, or the amount of water in a tank. It may be less appropriate for a ticket price, a shipping fee, or a digital counter, where jumps are normal.
The Intermediate Value Theorem gives a powerful existence result for a continuous function on an interval. If its output begins below a chosen level and ends above it, the graph must reach that level somewhere between. This helps locate roots by checking signs at endpoints.
It does not identify the exact location or guarantee only one solution. Later, derivatives rely on this nearby behavior to describe instantaneous rates and slopes.