Proving the Limit of x*sin(x) as x Approaches Infinity is Equal to 1

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Homework Help Overview

The problem involves evaluating limits, specifically focusing on the behavior of a function \( f(x) \) as \( x \) approaches infinity. The original poster is tasked with showing that the limit of \( \frac{f(x)}{x} \) equals 5, given a condition involving another limit that includes \( 5x^2 \sin(x) \).

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to manipulate the given limit condition to express \( f(x) \) and analyze its behavior as \( x \) approaches infinity. Some participants question the nature of \( f(x) \) and its implications for the limit, while others suggest that the original poster may be missing critical information from the problem statement.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and the implications of the limit condition. There is no explicit consensus on the nature of \( f(x) \) or the approach to take, but some guidance has been offered regarding the need for clarity in the problem's requirements.

Contextual Notes

There is uncertainty regarding the definition of \( f(x) \) and whether additional information is needed to proceed with the limit evaluation. The original poster acknowledges a potential omission in the problem statement.

karkas
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Homework Statement


The complete exercise is:

If [itex]\lim_{x->\inf } \frac{f(x)-5x^2sin(x)}{(\sqrt (x^2+2))-x} = 7[/itex]

show that [itex]\lim_{x->\inf} \frac{f(x)}{x} = 5[/itex]

Homework Equations


How do I show that [itex]\lim_{x->\inf} xsinx =1[/itex], because I run into it!

The Attempt at a Solution



I set K(x) = the fraction of the first limit and I solved for f(x) (x=0 excluded).

Then I have the limit [itex]\lim_{x->\inf} \frac{f(x)}{x} = \lim_{x->\inf} K(x)*0 + 5 xsinx[/itex].

Yet finally I reach the limit I spoke about in 2.
 
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You can't, it eventually oscillates between +/- infinity. What exactly is f(x) in this context?
 
Random function. It doesn't specify... Any other solutions?
 
Maybe you're omitting part of the question?
If [itex] \lim_{x->\inf } \frac{f(x)-5x^2sin(x)}{(\sqrt (x^2+2))-x}[/itex]
doesn't say anything because you're only giving the condition. Does the limit = something? Is the question asking you to find f(x) such that [itex]\lim_{x->\inf} \frac{f(x)}{x} = 5[/itex]?
 
Yes indeed I'll fix it.

No, it just wants me to prove the second limit equals 5.
 

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