Differential Calculus: Limits and Continuity
Limits and Continuity tests whether you can evaluate a limit correctly and then use the limit to decide continuity at a point, including piecewise-defined functions. Both one-sided limits and standard trigonometric and exponential limits appear.
Limit rules
Limits respect addition, subtraction, multiplication and division, provided the limit of the divisor is not zero. Standard results to have ready: lim (sin x)/x = 1, lim (tan x)/x = 1, lim (1 − cos x)/x² = ½, lim (eˣ − 1)/x = 1, lim log(1 + x)/x = 1, and lim (1 + x)^{1/x} = e, all as x → 0. A 0/0 or ∞/∞ form is indeterminate, so factorise, rationalise, or divide numerator and denominator by the highest power before substituting.
Continuity
A function f is continuous at x = a when the left-hand limit, the right-hand limit and f(a) all exist and are equal. All three conditions must hold.
Worked example
Evaluate lim as x → 0 of sin 3x/(5x). Rewrite it as (3/5) · (sin 3x)/(3x). Since 3x → 0, the bracket tends to 1, so the limit is 3/5.
Traps
- Substituting x = a before simplifying an indeterminate form.
- Ignoring one-sided limits at a break in a piecewise function.
- Writing 1/0 as infinity rather than recognising the limit does not exist.
- Confusing a removable discontinuity with continuity.