SUMMARY
The limit $\displaystyle \lim_{x\to 0} \frac{x^2-\sin^2{x}}{\tan(3x^4)}$ can be effectively calculated using L'Hôpital's Rule and series expansion techniques. Initially, applying L'Hôpital's Rule reveals a repeated $\displaystyle \frac{0}{0}$ indeterminate form, necessitating multiple iterations. Ultimately, the limit converges to $\frac{1}{9}$ through a series of transformations, including the use of trigonometric identities and simplifications.
PREREQUISITES
- Understanding of L'Hôpital's Rule for evaluating limits
- Familiarity with Taylor series expansions for trigonometric functions
- Knowledge of indeterminate forms in calculus
- Basic proficiency in manipulating limits involving trigonometric functions
NEXT STEPS
- Study advanced applications of L'Hôpital's Rule in calculus
- Learn about Taylor series expansions for $\sin x$ and $\tan x$
- Explore other methods for evaluating limits, such as the Squeeze Theorem
- Investigate the behavior of limits involving higher-order polynomials and trigonometric functions
USEFUL FOR
Students and professionals in mathematics, particularly those studying calculus and limit evaluation techniques, as well as educators looking for examples of complex limit calculations.
