Limits Questions with Solutions

The limit of a function describes the value the function approaches as its input approaches a given point. Limits are the foundation of calculus, because continuity, derivatives and integrals are all defined using them. This topic covers left-hand and right-hand limits, the algebra of limits, and standard methods of evaluation: direct substitution, factorisation, rationalisation, and dividing by the highest power for limits at infinity. It also covers the standard limits, such as (sin x)/x -> 1 as x -> 0, (e^x - 1)/x -> 1, (a^x - 1)/x -> log a, and (1 + x)^(1/x) -> e. Most exam questions are 0/0 forms that reduce to one of these after simplification. A limit exists only when the left-hand and right-hand limits are equal, which is especially important for the modulus and greatest integer functions. Practise the limits questions below, each with a step-by-step solution.