Limits and continuity · Year 13

In Year 13 you learn: limits.

A limit is the value a function gets closer and closer to as x approaches a number (or grows without bound). The function does not even need to be defined there. Limits are how calculus makes sense of instantaneous rates of change: every derivative is a limit.

Key facts

• The limit of f(x) as x → a is L if f(x) gets as close to L as we like when x is close enough to a (but not equal to a).
• If substituting gives 0/0, simplify first. Factorise and cancel, or rationalise with a conjugate. Then substitute.
• As x → ∞, divide top and bottom by the highest power of x. Terms like k/x tend to 0.
• Standard limits (x in radians): (sin x)/x → 1, (e^x − 1)/x → 1, (1 + k/n)^n → e^k.
• L’Hôpital’s rule: if f(a) = g(a) = 0, then lim f/g = lim (f′)/(g′) (when that limit exists).