Limits

Learn how to describe a limit as the value a function approaches near a point, write it in limit notation, distinguish the limit from $f(a)$, and recognise when a limit fails to exist.

Learn how to estimate a limit numerically by building a table that closes in on the target from both sides, and recognise when the table is unreliable.

Learn how to read a limit from a graph by tracing the curve toward a point, including holes and closed dots where the limit differs from $f(a)$, and recognise the three ways a limit fails to exist.

Learn how to state and apply the constant and identity limits and the sum, difference, and constant-multiple limit laws, then chain them to decompose a polynomial limit step by step into a number.

Learn how to state and apply the product and quotient limit laws to evaluate limits of products and quotients.

Learn how to state and apply the power and root limit laws to evaluate limits of powers and roots.

Learn how to decompose a compound limit by working from the outermost operation inwards, applying the matching law at each step and checking its condition where one applies, until only basic limits remain.

Learn how to evaluate a limit by direct substitution when its conditions hold, giving $\lim\limits_{x \to a} f(x) = f(a)$, and recognise when substitution fails as the indeterminate form $0/0$ or the non-existent form $c/0$.

Learn how to resolve a limit that direct substitution leaves as $0/0$, either by factoring and cancelling a shared factor or by rationalising a square root with its conjugate, then substituting into the simplified form.

Learn how to compute the one-sided limits $\lim\limits_{x \to a^-} f$ and $\lim\limits_{x \to a^+} f$ by evaluating the branch that applies on each side, then decide whether the two-sided limit exists by checking whether the two sides agree.

Learn how to evaluate infinite one-sided limits by reading the sign of the function near the point, and how to locate the vertical asymptotes of a rational function at the denominator zeros that do not cancel.

Learn how to evaluate limits as $x \to \pm\infty$, reading a polynomial's end behaviour off its leading term and a rational function's limit by dividing by the highest power of $x$ and comparing the degrees of the numerator and denominator.

Learn how to evaluate limits of exponentials and logarithms, reading the outcome from the sign of the constant in the exponent, and why a logarithm grows without bound even though its graph flattens.

Learn how to find a function's horizontal asymptotes from its limits as $x \to \pm\infty$, and why a graph may cross an asymptote that only describes its end behaviour.

Learn how to test whether a function is continuous at a point using its three conditions, classify a discontinuity as removable, jump or infinite by which condition fails, and decide when a composition of continuous functions is continuous.

Learn how to repair a removable discontinuity by defining the function to equal its limit at the hole, and how to choose a parameter that makes a piecewise function continuous by matching its rules at a boundary.

State the Intermediate Value Theorem, check its hypotheses on a closed interval, and apply it to guarantee that a value or a root exists between two endpoints of opposite sign.

Learn how to recognise when a quotient limit is indeterminate, and how to resolve it with L'Hôpital's Rule by differentiating the numerator and the denominator separately and classifying again.