Limit laws are rules that let you find complicated limits by breaking them into simpler pieces. They are the main algebra toolkit for early calculus because they connect limits to familiar operations like adding, multiplying, dividing, and taking powers. When the separate pieces of a function have limits, these laws often let you evaluate the whole limit without a table or graph.
This makes limits faster, clearer, and less dependent on guessing from nearby values.
The basic idea is that if lim as x approaches a of f(x) and lim as x approaches a of g(x) both exist, then many combinations of f and g also have predictable limits. For example, the limit of a sum is the sum of the limits, and the limit of a product is the product of the limits. The quotient law requires the denominator limit to be nonzero, and root laws may require domain restrictions for even roots.
In practice, students use limit laws to substitute x = a when the function is continuous, or to simplify first when direct substitution gives an indeterminate form such as 0/0.
Key Facts
- Sum law: If lim f(x) = L and lim g(x) = M, then lim [f(x) + g(x)] = L + M.
- Difference and constant multiple laws: lim [f(x) - g(x)] = L - M and lim [c f(x)] = cL.
- Product law: If lim f(x) = L and lim g(x) = M, then lim [f(x)g(x)] = LM.
- Quotient law: lim [f(x)/g(x)] = L/M, provided M is not 0.
- Power law: lim [f(x)]^n = L^n for positive integer n, and more generally for valid real powers when the expression is defined near a.
- Root and composition laws: lim n√f(x) = n√L when the root is defined, and if g is continuous at L, then lim g(f(x)) = g(L).
Vocabulary
- Limit
- A limit is the value that a function approaches as the input gets closer to a chosen number.
- Limit law
- A limit law is a rule that allows limits of combined functions to be found from the limits of their parts.
- Continuity
- A function is continuous at a point when its limit there equals its actual function value.
- Indeterminate form
- An indeterminate form is an expression such as 0/0 that does not reveal the limit without further simplification.
- Composition
- A composition is a function made by using the output of one function as the input of another, written g(f(x)).
Common Mistakes to Avoid
- Using the quotient law when the denominator limit is 0 is wrong because division by zero is not defined and the limit may need simplification or may not exist.
- Substituting x = a after getting 0/0 and stopping is wrong because 0/0 is indeterminate, not proof that the limit is 0 or undefined.
- Applying root laws without checking the domain is wrong because even roots require nonnegative inputs near the point in the real number system.
- Assuming every limit can be split apart is wrong because limit laws require the component limits to exist and the expressions to be defined under the law being used.
Practice Questions
- 1 Evaluate lim as x approaches 2 of (3x^2 - 5x + 4) using limit laws.
- 2 Evaluate lim as x approaches 1 of (x^2 - 1)/(x - 1) by simplifying first, then applying limit laws.
- 3 Explain why the quotient law cannot be directly used to evaluate lim as x approaches 0 of sin(x)/x, even though the limit exists.