Studying a function using comparative growth rates

Objective

Comparative growth is often used to resolve indeterminate forms when studying a function: limits at bounds, finding asymptotes, and behaviour as the variable approaches +∞ or 0+.

General method

When dealing with an expression combining polynomials, exponential functions and logarithms:

1. Identify the indeterminate form (∞ – ∞, ∞/∞, 0 × ∞, etc.).
2. Factorise using the dominant term, identified using the hierarchy ln(x) << x^n << e^x.
3. Apply the comparative growth theorems to the residual terms, which tend to 0.

Example 1: analysis at +∞

Let f(x) = x² * e^(-x). At +∞, we write f(x) = x²/e^x, a form of the type ∞/∞. According to the theorem, $x^2/e^x \to 0$, so $f$ has a horizontal asymptote of equation $y = 0$ at $+\infty$.

Example 2: Analysis at 0+

Let g(x) = x * ln(x) - x, for x → 0+. We have x*ln(x) → 0 (comparative growth, case n=1) and -x → 0, so g(x) → 0. Note: this is not sufficient to claim that g is continuous at 0 without explicitly specifying an extension such that g(0)=0.

Pitfall to watch out for

ln(x) is only defined for x > 0. Before applying a comparative growth theorem in the neighbourhood of 0, check that the function is indeed defined to the right of this point.

Summary table of forms to recognise

Form encountered	Dominant term	Limit
x^n * e^(-x), x → +∞	e^x	0
x^n * ln(x), x → 0+	1/x^n (indirectly)	0
(ln x)/x^n, x → +∞	x^n	0