SUMMARY
The limit of the expression \(\frac{log (x^{2} + e^{2x})}{x + 3}\) as \(x\) approaches infinity is determined using the concept that exponential functions dominate logarithmic functions. Specifically, the term \(e^{2x}\) in the numerator significantly outweighs \(x^{2}\) for large \(x\). By applying the sandwich theorem, it is established that \(\frac{log (e^{2x})}{x + 3}\) approaches 2, confirming that the limit is indeed 2 as \(x\) tends to infinity.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with logarithmic and exponential functions
- Knowledge of the sandwich theorem
- Basic algebraic manipulation skills
NEXT STEPS
- Study the properties of logarithmic and exponential functions
- Learn about the sandwich theorem in detail
- Explore advanced limit techniques in calculus
- Practice solving limits involving logarithmic and exponential expressions
USEFUL FOR
Students studying calculus, particularly those focusing on limits, as well as educators looking for examples of applying the sandwich theorem and comparing growth rates of functions.