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 fx is b as x approaches a, and the limit of gx is c as x approaches b, then the limit of gf(x) as x approaches a is c.
Example: If lim fx = b as x→a, and lim gx = c as x→b, then lim gf(x) = c as x→a
Comparison Method:
This section introduces techniques for evaluating limits by comparing functions:
- If the limit of fx is positive infinity, and fx ≤ gx for x sufficiently large, then the limit of gx is also positive infinity.
- If the limit of gx is 0, and |fx| ≤ gx for x sufficiently large, then the limit of fx 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 gx ≤ fx ≤ hx for all x in some interval containing a exceptpossiblyataitself, and the limits of gx and hx both equal L as x approaches a, then the limit of fx 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.
