Find the limit in question 7e and 7g?

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SUMMARY

The limits in questions 7e and 7g can be determined using calculus techniques. For question 7e, the limit evaluates to zero, as established by the inequality 0 ≤ |x sin(1/x)| ≤ |x|, which approaches zero as x approaches zero. In question 7g, the limit evaluates to one by expanding sin(1/x) around x approaching infinity, resulting in the expression x sin(1/x) simplifying to x [1/x + O(1/x^3)], which converges to one.

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How can I find the limit in question 7e and 7g?
Finding limits is difficult!:cry:
 

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7e: I guess everything is within real calculus. Then you can argue that the sine is bound and thus the limit must be zero:
[tex]0 \leq |x \sin(1/x)| \leq |x| \rightarrow 0.[/tex]

7g: Expanding [itex]\sin(1/x)[/itex] around [itex]x \rightarrow \infty[/itex] gives
[tex]x \sin(1/x)=x [1/x+\mathcal{O}(1/x^3)] \rightarrow 1.[/tex]
 
vanhees71 said:
7e: I guess everything is within real calculus. Then you can argue that the sine is bound and thus the limit must be zero:
[tex]0 \leq |x \sin(1/x)| \leq |x| \rightarrow 0.[/tex]

7g: Expanding [itex]\sin(1/x)[/itex] around [itex]x \rightarrow \infty[/itex] gives
[tex]x \sin(1/x)=x [1/x+\mathcal{O}(1/x^3)] \rightarrow 1.[/tex]

How to derive this?
Could you explain more,please?
 

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