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player1_1_1
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Homework Statement
hello, what can I do to solve this without using d'hospital method?
\lim_{x\to0}\frac{\tan x-x}{x^3}
How can I solve this limit without using d'hospital method?
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SUMMARY
The limit \(\lim_{x\to0}\frac{\tan x-x}{x^3}\) can be solved without using L'Hôpital's rule by employing the Taylor series expansion for \(\tan(x)\). The series is given by \(\tan(x) = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + \frac{17}{315}x^7 + \ldots\). By substituting this expansion into the limit expression, the \(x\) terms cancel, allowing for simplification. After dividing both the numerator and denominator by \(x^3\) and applying direct substitution, the limit evaluates to \(\frac{1}{3}\).
PREREQUISITES- Taylor series expansion for trigonometric functions
- Understanding of limits in calculus
- Basic algebraic manipulation skills
- Familiarity with the concept of indeterminate forms
- Study the Taylor series for \(\sin(x)\) and \(\cos(x)\)
- Learn about indeterminate forms and their resolutions
- Explore alternative methods for evaluating limits
- Practice solving limits using series expansions
Students studying calculus, particularly those focusing on limits and series expansions, as well as educators seeking effective methods for teaching these concepts.