Limits of ln and comparative growth rates

Limits at the boundaries of the domain

The function ln has two fundamental limits that must be learnt by heart:

limit of ln(x) as x approaches 0 (x > 0) = negative infinity

limit of ln(x) as x tends to +∞ = +∞

The first limit reflects the existence of a vertical asymptote with equation x = 0 for the graph of ln.

Very slow growth

Although ln(x) tends towards +∞, it does so extremely slowly, much more slowly than x itself. This is known as a comparative growth:

limit of ln(x)/x as x tends towards +∞ = 0

limit of x × ln(x) as x tends to 0 (x > 0) = 0

These two results allow us to resolve indeterminate forms of the type ∞/∞ or 0 × ∞ in limit calculations.

Solving an equation involving ln

To solve ln(x) = k (where k is a given real number), we use the definition: x = e^k; this solution is strictly positive and therefore always valid.

Example: Solve ln(2x – 1) = 0. The existence condition requires that 2x – 1 > 0, i.e. x > 1/2. We then solve 2x – 1 = e⁰ = 1, so x = 1, which indeed satisfies the condition. The solution is x = 1.

Solving an inequality involving ln

Example: Solve ln(x) ≤ 2, for x > 0. As ln is strictly increasing, this is equivalent to x ≤ e² . Taking the domain into account, the set of solutions is ]0; e²].

Common pitfall

Always check the existence condition (the argument of ln must be strictly positive) BEFORE solving, and verify at the end that the solutions found do indeed satisfy this condition. Any solution that does not satisfy it must be rejected.