Concept / Calculus
A limit studies approach, not arrival.
The value of a function at a point and the behavior of nearby values are different questions. Limits provide the language for studying what a function approaches as its input moves toward a boundary or missing point.
The central distinction
This statement says that f(x) can be made as close to L as desired by taking x sufficiently close to a, while keeping x distinct from a. The expression does not require f(a) to equal L, and it does not even require f(a) to be defined.
Three values that must not be confused
| Quantity | Question |
|---|---|
| f(a) | What value, if any, is assigned at the point? |
| limx→a- f(x) | What is approached from values smaller than a? |
| limx→a+ f(x) | What is approached from values larger than a? |
A finite two-sided limit exists only when the left- and right-hand limits both exist and agree. A jump has two unequal finite one-sided limits. A vertical asymptote can produce infinite one-sided behavior. Oscillation may prevent either from settling.
A removable singularity
The formula is undefined at zero because it asks for division by zero. Nevertheless, nearby values approach 1 from both sides. Defining f(0)=1 fills the hole and creates a continuous extension.
The precise idea
For every ε greater than zero, there must be a δ greater than zero such that whenever 0 < |x-a| < δ, the output satisfies |f(x)-L| < ε. The excluded center, 0 < |x-a|, is what separates approach behavior from the point value.