A limit describes the output approached as the input approaches a location; the value at that location can be different or undefined. A two-sided finite limit exists only when both one-sided limits agree. Tables support a conjecture but do not prove behavior between entries. On a graph follow the branches, rather than the filled point. Interpret average rates over shrinking intervals as motivation for instantaneous change.
For continuous elementary expressions, substitution is efficient wherever the expression is defined. A 0/0 result signals an indeterminate form, not a value. Factor and cancel only on a punctured neighborhood, or multiply by a conjugate. The simplified expression has the same nearby values even when the original has a hole. Check left and right behavior if an absolute value or piecewise rule changes at the approach point.
If a function is trapped between two functions with the same limiting value, it shares that limit. Infinite limits describe unbounded output and require sign analysis on each side of a zero denominator. Limits at infinity describe end behavior. For rational functions divide by the dominant power or compare degrees; a horizontal asymptote may be crossed. Standard trigonometric limits use radians.
Continuity at an interior point requires a defined function value, an existing finite limit, and equality of that limit to the function value. At an included endpoint use the relevant one-sided limit. A removable discontinuity can be fixed by assigning the limiting value. Jumps and infinite discontinuities cannot be fixed by changing a single function value. For piecewise rules compare the appropriate formulas at every join.
A function continuous on a closed interval takes every value between its endpoint outputs. To prove a root exists, establish continuity and opposite endpoint signs. The theorem guarantees at least one solution, not uniqueness or its location. Graphs and tables help choose an interval, but the written argument must state why continuity holds.
