$Find $\lim_{x \rightarrow} x^2 \sin(1/x) $. Cite theorems used and show all work$

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$Find $\lim_{x \rightarrow} x^2 \sin(1/x) $. Cite theorems used and show all work$

Show all work and cite theorems used.

Calculus

Scuentiest134

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Since $-1\leq \sin(\frac{1}{x})\leq 1$, we have
\[-x^2\leq x^2 \sin (\frac{1}{x}) \leq x^2.\]
Also
\[\lim_{x \rightarrow 0} -x^2\leq \lim_{x \rightarrow 0} x^2 \sin (\frac{1}{x}) \leq \lim_{x \rightarrow 0} x^2.\]
Since $\lim_{x \rightarrow 0} -x^2=\lim_{x \rightarrow 0} x^2=0$, by the squeez theorem ( https://en.wikipedia.org/wiki/Squeeze_theorem ) we conclude that
\[0\leq \lim_{x \rightarrow 0} x^2 \sin (\frac{1}{x}) \leq 0.\]
Therefore
\[\lim_{x \rightarrow 0} x^2 \sin (\frac{1}{x})=0.\]

Savionf

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Savionf

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Please leave a comment if you need any clarifications.
Scuentiest134

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Thank you !
- Savionf
  
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  You're welcome. I am glad I was able to help.

$Find $\lim_{x \rightarrow} x^2 \sin(1/x) $. Cite theorems used and show all work$

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