Advanced Limit Techniques
This page delves into more sophisticated methods for evaluating limits, building upon the foundational concepts introduced earlier. These techniques are essential for tackling complex limit problems often found in exercice type BAC limite de fonction Terminale S pdf.
Composition of Functions:
The guide explains how to handle limits of composite functions. If the limit of f(x) is b as x approaches a, and the limit of g(x) is c as x approaches b, then the limit of g(f(x)) as x approaches a is c.
Example: If lim f(x) = b as x→a, and lim g(x) = c as x→b, then lim g(f(x)) = c as x→a
Comparison Method:
This section introduces techniques for evaluating limits by comparing functions:
- If the limit of f(x) is positive infinity, and f(x) ≤ g(x) for x sufficiently large, then the limit of g(x) is also positive infinity.
- If the limit of g(x) is 0, and |f(x)| ≤ g(x) for x sufficiently large, then the limit of f(x) is also 0.
Growth Rate Comparison:
The guide presents important results about the relative growth rates of exponential and polynomial functions:
- The limit of e^x / x^n as x approaches positive infinity is positive infinity for any positive integer n.
- The limit of x^m * e^(-x) as x approaches positive infinity is 0 for any real number m.
Highlight: These growth rate comparisons are crucial for solving limites et continuité exercices corrigés PDF.
Squeeze Theorem (Encadrement):
The final section introduces the squeeze theorem, a powerful tool for evaluating limits:
If g(x) ≤ f(x) ≤ h(x) for all x in some interval containing a (except possibly at a itself), and the limits of g(x) and h(x) both equal L as x approaches a, then the limit of f(x) as x approaches a also equals L.
Definition: The squeeze theorem allows us to determine the limit of a function by "squeezing" it between two functions with known limits.
This advanced technique is particularly useful for solving complex limit problems and is often featured in exercices corrigés limites fonctions composées.
