Notable limits and comparative growth rates

Important limits to know off by heart

lim(x→0) sin(x)/x = 1 lim(x→0) (1-cos(x))/x² = 1/2 lim(x→0) (e^x - 1)/x = 1 lim(x→0) ln(1+x)/x = 1 lim(x→+∞) (1+1/x)^x = e

These limits serve as the building blocks for calculating others, through composition or a change of variable.

Equivalence at a point

Two functions f and g are equivalent at a (denoted f ~ g) if lim(x→a) f(x)/g(x) = 1. We can then replace f with a simpler equivalent g in a product or a quotient, but never in a sum or a difference.

Example: sin(x) ~ x at 0, so lim(x->0) sin(3x)/(5x) = lim(x->0) 3x/(5x) = 3/5.

Comparing growth rates as x approaches +∞

For any a > 0 and b > 1, we classify the reference functions as follows:

Ascending order	Function
1 (slowest)	ln(x)
2	x^a
3	b^x
4 (fastest)	x^x

This is often summarised as: ln(x) << x^a << b^x as x approaches positive infinity. The logarithm is ‘dominated’ by any power, which is itself dominated by any exponential function with a base greater than 1.

Common pitfall

Using an equivalent in a sum or difference f – g when f ≈ g: this may yield 0 instead of the correct result. It is then necessary to resort to a higher-order limit expansion to capture the actual dominant term.