Pregunta
44. \( \lim _{x \rightarrow 0^{+}} x^{3} \cot x \)
Ask by Kelly Craig. in the United States
Jan 23,2025
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The limit is 0.
Solución
Calculate the limit \( \lim_{x \rightarrow 0^{+}} x^{3} \cot x \).
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
\(\lim _{x\rightarrow 0^{+}}\left(x^{3}\cot\left(x\right)\right)\)
- step1: Transform the expression:
\(\lim _{x\rightarrow 0^{+}}\left(\frac{x^{3}}{\tan\left(x\right)}\right)\)
- step2: Use the L'Hopital's rule:
\(\lim _{x\rightarrow 0^{+}}\left(\frac{\frac{d}{dx}\left(x^{3}\right)}{\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)\)
- step3: Find the derivative:
\(\lim _{x\rightarrow 0^{+}}\left(\frac{3x^{2}}{\sec^{2}\left(x\right)}\right)\)
- step4: Rewrite the expression:
\(\frac{\lim _{x\rightarrow 0^{+}}\left(3x^{2}\right)}{\lim _{x\rightarrow 0^{+}}\left(\sec^{2}\left(x\right)\right)}\)
- step5: Calculate:
\(\frac{0}{\lim _{x\rightarrow 0^{+}}\left(\sec^{2}\left(x\right)\right)}\)
- step6: Calculate:
\(\frac{0}{1}\)
- step7: Calculate:
\(0\)
The limit of \(x^3 \cot x\) as \(x\) approaches 0 from the right is 0.
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Simplifique esta solución Bonus Knowledge
To find \( \lim_{x \rightarrow 0^{+}} x^{3} \cot x \), we can rewrite \(\cot x\) as \(\frac{\cos x}{\sin x}\). Thus, the limit becomes: \[ \lim_{x \rightarrow 0^{+}} x^{3} \frac{\cos x}{\sin x} \] As \(x\) approaches \(0\), \(\cos x\) approaches \(1\), and \(\sin x\) can be approximated by its Taylor series expansion, which gives \(\sin x \approx x\) near zero. Therefore, we can substitute: \[ \lim_{x \rightarrow 0^{+}} x^{3} \frac{1}{\sin x} = \lim_{x \rightarrow 0^{+}} x^{3} \frac{1}{x} = \lim_{x \rightarrow 0^{+}} x^{2} = 0 \] Thus, the limit is: \[ \boxed{0} \]
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