September 19th, 2018, 08:12 PM  #1 
Senior Member Joined: Apr 2017 From: New York Posts: 119 Thanks: 6  limit convergent
]How can I approach this question, please?

September 19th, 2018, 08:53 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,515 Thanks: 2515 Math Focus: Mainly analysis and algebra 
Notice that as $y \to 0^+$ the expression $\frac1{y^b}$ grows without bound for all $b \gt 0$. In this case the sine function oscillates more and more rapidly, so you need the other term to shrink the oscillations to zero. When $b = 0$, the sine function is constant. You can consider what that means for the other term. When $b < 0$, $\frac1{y^b} \to 0$ and so you will need to consider that $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ By setting $x = \frac1{y^b}$, you will be able to reduce the given limit to one that only involves powers of $y$ and thus solve the problem Last edited by v8archie; September 19th, 2018 at 08:58 PM. 
September 19th, 2018, 11:23 PM  #3 
Senior Member Joined: Apr 2017 From: New York Posts: 119 Thanks: 6 
So to be able to graph on a and b axes what values should I give for a and b or how can I draw it? 
September 19th, 2018, 11:38 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,515 Thanks: 2515 Math Focus: Mainly analysis and algebra 
For each of those three cases you should be able to determine the boundary between convergence and divergence in the form of a curve. For example if it is required that $a<b$, the boundary is $a=b$. You can draw this and shade the appropriate side.

September 20th, 2018, 06:31 AM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
sinx$\displaystyle \le$1, all x (all b) let y = 1/n lim 1/y$\displaystyle ^{a}$ = lim n$\displaystyle ^{a}$ = $\displaystyle \infty$ for a>0, = 0 for all a $\displaystyle \le$ 0 
September 20th, 2018, 09:11 AM  #6  
Senior Member Joined: Sep 2016 From: USA Posts: 522 Thanks: 297 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
\[ \sin(x) = x\cos(\theta x + 1  \theta) \ \text{for some } \theta \in (0,1) \] Evaluate at $y^{b}$, multiply by $y^{a}$ and let $y \to 0$.  
September 20th, 2018, 12:28 PM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  For a=1, 1/y$\displaystyle ^{a}$ = y whose limit is zero (given). Your conclusion that I was not correct based on an incorrect counterexample is wrong. Presumably (hopefully) OP will eventually learn correct answer in class. Considering his judgement in awarding thanks, I would appreciate it if he would remove mine. 
September 20th, 2018, 02:30 PM  #8 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  
September 20th, 2018, 03:59 PM  #9  
Senior Member Joined: Sep 2016 From: USA Posts: 522 Thanks: 297 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 
September 20th, 2018, 05:31 PM  #10 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  

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