Limit of a Function
The limit of f at a is L when f(x) can be made arbitrarily close to L by taking x sufficiently close to a but not equal to a.
What is Limit of a Function?
The limit of f at a is L when f(x) can be made arbitrarily close to L by taking x sufficiently close to a but not equal to a.
Key formula / rule: A limit exists only when the left and right limits agree.
Key points
- Define the limit of a function and test existence using one-sided limits.
- Compute limits using algebraic simplification and standard limits.
Common exam trap
Substituting the point directly into an expression of the form 0/0.
Definitions
- Term
Limit of a Function
- Meaning
The limit of f at a is L when f(x) can be made arbitrarily close to L by taking x sufficiently close to a but not equal to a.
- Term
Limit of a Function — explanation
- Meaning
The limit describes the value a function approaches, deliberately ignoring what happens at the point itself. That separation is what allows derivatives, which are limits of quotients that are undefined at the point.
Learning objectives
Define the limit of a function and test existence using one-sided limits.
Compute limits using algebraic simplification and standard limits.
Formulae
- Key point
A limit exists only when the left and right limits agree.
- Key point
The value f(a) is irrelevant to the existence of the limit at a.
- Key point
Algebraic limit laws apply only when the individual limits exist finitely.
Common mistakes
Substituting the point directly into an expression of the form 0/0.
Assuming a two-sided limit exists when only one side has been checked.
Keywords
Limit
Function
Practice preview
Evaluate the limit: lim (x->2) (x^2 + 3x - 5).…
easy
Evaluate the limit: lim (x->0) (sin(4x)) / x.…
easy
Evaluate the limit: lim (x->0) (sqrt(1+x) - 1) / x.…
medium