To find the limit as [tex]\( x \)[/tex] approaches 0 for the expression [tex]\( \frac{\tan(5x) - \sin(5x)}{x^3} \)[/tex], we need to carefully analyze the behavior of both the tangent and sine functions near zero. Here's a detailed, step-by-step solution:
1. Expression Reformulation:
Given the limit:
[tex]\[
\lim_{x \to 0} \frac{\tan(5x) - \sin(5x)}{x^3}
\][/tex]
2. Taylor Series Expansions:
We know the Taylor series expansions for [tex]\( \tan(5x) \)[/tex] and [tex]\( \sin(5x) \)[/tex] at [tex]\( x = 0 \)[/tex] are:
[tex]\[
\tan(5x) = 5x + \frac{(5x)^3}{3} + O(x^5)
\][/tex]
[tex]\[
\sin(5x) = 5x - \frac{(5x)^3}{6} + O(x^5)
\][/tex]
where [tex]\( O(x^5) \)[/tex] denotes higher-order terms which become very small as [tex]\( x \)[/tex] approaches 0.
3. Subtracting the Series:
Subtract the series expansion of [tex]\( \sin(5x) \)[/tex] from [tex]\( \tan(5x) \)[/tex]:
[tex]\[
\tan(5x) - \sin(5x) = \left(5x + \frac{(5x)^3}{3} + O(x^5)\right) - \left(5x - \frac{(5x)^3}{6} + O(x^5)\right)
\][/tex]
Simplify the subtraction:
[tex]\[
\tan(5x) - \sin(5x) = \frac{(5x)^3}{3} + \frac{(5x)^3}{6} + O(x^5)
\][/tex]
4. Combining Like Terms:
Combine the like terms from the cubic part:
[tex]\[
\tan(5x) - \sin(5x) = \frac{125x^3}{6} + O(x^5)
\][/tex]
5. Forming the Limit Expression:
Substitute this back into the original limit:
[tex]\[
\lim_{x \to 0} \frac{\frac{125x^3}{6} + O(x^5)}{x^3}
\][/tex]
6. Simplifying the Limit Expression:
Simplify the fraction inside the limit:
[tex]\[
\lim_{x \to 0} \left( \frac{125x^3}{6x^3} + \frac{O(x^5)}{x^3} \right)
\][/tex]
[tex]\[
\lim_{x \to 0} \left( \frac{125}{6} + O(x^2) \right)
\][/tex]
7. Evaluating the Limit:
As [tex]\( x \)[/tex] approaches 0, the higher-order term [tex]\( O(x^2) \)[/tex] approaches 0. Thus:
[tex]\[
\lim_{x \to 0} \left( \frac{125}{6} + 0 \right) = \frac{125}{6}
\][/tex]
8. Final Answer:
[tex]\[
\frac{125}{6} = \frac{125}{2}
\][/tex]
Therefore, the limit as [tex]\( x \)[/tex] approaches 0 for the expression [tex]\( \frac{\tan(5x) - \sin(5x)}{x^3} \)[/tex] is [tex]\( \frac{125}{2} \)[/tex].
