Each card is a single, complete idea: what it is, the formulas you need, and one worked example. Built to be re-read.
A limit is the value a function approaches as x gets infinitely close to a certain value. It describes the function's behavior near a point, not necessarily the value at that point.
Approach the point from the left (x→c−) and from the right (x→c+). The general (two-sided) limit exists only when both one-sided limits exist and are equal.
The limit is about the approach; f(c) is about the actual output. They can disagree at holes and at redefined points of piecewise functions.
Read limits from a graph by tracing each side toward the target x-value, or from a table by watching the numerical trend from both directions (round to three decimal places).
Try direct substitution first. If it gives 00, manipulate the expression — factor and cancel, combine fractions, or multiply by the conjugate — then substitute again.
Limits distribute over sums, differences, products, quotients (nonzero denominator), and constant multiples, which lets you break a messy limit into pieces.
Two identities handle most trig limits as the input approaches zero. Rewrite the expression so the angle inside matches the denominator.
For rational functions, compare degrees of numerator and denominator to find horizontal asymptotes. Divide by the highest power of x when in doubt.
If a function is trapped between two others that converge to the same value at a point, the trapped function must converge to that value too.
f is continuous at x=c only if all three hold: f(c) is defined, limx→cf(x) exists, and the limit equals f(c). Check them in order and stop at the first failure.
Removable holes come from factors that cancel; vertical asymptotes come from factors left in the denominator; jumps come from piecewise pieces whose one-sided limits disagree.
Polynomial, rational, exponential, logarithmic, and trig functions are continuous on their domains. For piecewise functions, force the pieces to agree at each boundary.
If f is continuous on [a,b], then f takes on every value K between f(a) and f(b) at least once. Always state continuity and that K lies between f(a) and f(b).