Limits and continuity: reasoning about approaching

The statement that $f(x)$ approaches $L$ as $x$ approaches $a$ means that outputs can be forced as close to $L$ as desired by restricting inputs sufficiently close to $a$, while excluding $x=a$ itself. The output tolerance is chosen first; the required input tolerance may depend on it. This is a claim about every sufficiently close allowed input, not just a few entries in a table.

A two-sided limit requires agreement from both sides, whenever the domain has nearby points on each side. If the left side approaches one number and the right side another, there is no common two-sided limit. A finite plot can suggest a limit but cannot establish all arbitrarily small scales. Oscillation, steep variation, or numerical rounding can hide behavior that matters near the point.

The expression $(x^2-1)/(x-1)$ is undefined at $x=1$. Factoring its numerator gives $(x-1)(x+1)$, so for every other input it equals $x+1$. Canceling the factor does not retroactively define the original expression at one. It instead gives a simpler rule with the same nearby behavior, which is exactly the information a limit asks for.

This is a removable discontinuity: assign the missing value to the nearby limit and the resulting extension becomes continuous. Other failures cannot be repaired by changing a single point. A jump has incompatible one-sided limits, and an unbounded function near a point has no finite limiting value there. Identify the kind of behavior before applying a memorized simplification rule.

Continuity at an interior point requires that the function be defined there, that its finite limit exist there, and that the two agree. Polynomials satisfy these conditions at every real input. Quotients of continuous functions do so wherever the denominator is nonzero. At an endpoint of a domain, continuity uses the appropriate one-sided approach rather than demanding inputs outside the domain.

Continuity is a property of the stated model, not a guarantee that it is physically accurate or well conditioned. A continuous function can change very quickly, and small input errors can still create significant output errors. Conversely, a digital decision rule can deliberately jump at a threshold. Ask whether a continuous approximation suits the question instead of treating continuity as an automatic mark of quality.