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for which values of parameter $a$ and $c$ function

0

$begingroup$

For which values of parameter $a$ and $c$ function

$$ f(x)= left{ beginarrayll
|x|^asin|x|^-c & textrmfor $x neq 0$\
0 & textrmfor $x=0$
endarray right.
$$

a) is continuous on the interval $[-1,1]$

b) it is differentiable in $[-1,1]$

c) the derivative is limited

this is my homework.
i must to calculate $lim_x to 0^- $and $lim_xto 0^+$ this is the same lim.

So $lim_x to 0^-|x|^asin|x|^-c$

I don't know how calculate this...
and what next b) and c) ..

real-analysis functions

share|cite|improve this question

edited yesterday

Mostafa Ayaz

18.1k31040

asked yesterday

mona1lisamona1lisa

727

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it $sin left( left| x right|^-c right)$ or $left( sin left| x right| right)^-c$?
  $endgroup$
  – Aniruddha Deshmukh
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  there are no parentheses in the notebook
  $endgroup$
  – mona1lisa
  yesterday
  

  
  
  
  

add a comment | 

0

$begingroup$

For which values of parameter $a$ and $c$ function

$$ f(x)= left{ beginarrayll
|x|^asin|x|^-c & textrmfor $x neq 0$\
0 & textrmfor $x=0$
endarray right.
$$

a) is continuous on the interval $[-1,1]$

b) it is differentiable in $[-1,1]$

c) the derivative is limited

this is my homework.
i must to calculate $lim_x to 0^- $and $lim_xto 0^+$ this is the same lim.

So $lim_x to 0^-|x|^asin|x|^-c$

I don't know how calculate this...
and what next b) and c) ..

real-analysis functions

share|cite|improve this question

edited yesterday

Mostafa Ayaz

18.1k31040

asked yesterday

mona1lisamona1lisa

727

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it $sin left( left| x right|^-c right)$ or $left( sin left| x right| right)^-c$?
  $endgroup$
  – Aniruddha Deshmukh
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  there are no parentheses in the notebook
  $endgroup$
  – mona1lisa
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

For which values of parameter $a$ and $c$ function

$$ f(x)= left{ beginarrayll
|x|^asin|x|^-c & textrmfor $x neq 0$\
0 & textrmfor $x=0$
endarray right.
$$

a) is continuous on the interval $[-1,1]$

b) it is differentiable in $[-1,1]$

c) the derivative is limited

this is my homework.
i must to calculate $lim_x to 0^- $and $lim_xto 0^+$ this is the same lim.

So $lim_x to 0^-|x|^asin|x|^-c$

I don't know how calculate this...
and what next b) and c) ..

real-analysis functions

share|cite|improve this question

edited yesterday

Mostafa Ayaz

18.1k31040

asked yesterday

mona1lisamona1lisa

727

$endgroup$

share|cite|improve this question

edited yesterday

Mostafa Ayaz

18.1k31040

asked yesterday

mona1lisamona1lisa

727

share|cite|improve this question

edited yesterday

Mostafa Ayaz

18.1k31040

asked yesterday

mona1lisamona1lisa

727

edited yesterday

Mostafa Ayaz

18.1k31040

edited yesterday

Mostafa Ayaz

18.1k31040

asked yesterday

mona1lisamona1lisa

727

asked yesterday

mona1lisamona1lisa

727

1 Answer
1

active

oldest

votes

0

$begingroup$

I assume the functions is $$f(x)=|x|^asin(|x|^-c)$$over $Bbb R-0$. Note that this function is even, therefore$$lim_xto 0f(x)=lim_xto 0^+f(x)$$Now, we consider $3$ cases (in al of the cases we consider $xne 0$):

Case 1: $a>0$

The function is continuous by Sqeeze theorem since $$-|x|^a<f(x)<|x|^a$$

Case 2: $a=0$

The function is continuous only if $c<0$.

Case 3: $a<0$

In this case, the function has no limit at $x=0$ when $c=0$ or $c>0$. For $c<0$ we can write:$$lim_xto 0f(x)^-c)\=lim_xto 0fracx^-a\=lim_xto 0fracx^-ccdotx\=lim_xto 0^a-c$$which is equal to zero only if $a>c$.

Conclusion

The values of $a,c$ for which $f(x)$ is continuous is as follows:$$a>0\a=0,c<0\c<a<0$$

share|cite|improve this answer

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

add a comment | 

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0

$begingroup$

I assume the functions is $$f(x)=|x|^asin(|x|^-c)$$over $Bbb R-0$. Note that this function is even, therefore$$lim_xto 0f(x)=lim_xto 0^+f(x)$$Now, we consider $3$ cases (in al of the cases we consider $xne 0$):

Case 1: $a>0$

The function is continuous by Sqeeze theorem since $$-|x|^a<f(x)<|x|^a$$

Case 2: $a=0$

The function is continuous only if $c<0$.

Case 3: $a<0$

In this case, the function has no limit at $x=0$ when $c=0$ or $c>0$. For $c<0$ we can write:$$lim_xto 0f(x)^-c)\=lim_xto 0fracx^-a\=lim_xto 0fracx^-ccdotx\=lim_xto 0^a-c$$which is equal to zero only if $a>c$.

Conclusion

The values of $a,c$ for which $f(x)$ is continuous is as follows:$$a>0\a=0,c<0\c<a<0$$

share|cite|improve this answer

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

add a comment | 

0

$begingroup$

I assume the functions is $$f(x)=|x|^asin(|x|^-c)$$over $Bbb R-0$. Note that this function is even, therefore$$lim_xto 0f(x)=lim_xto 0^+f(x)$$Now, we consider $3$ cases (in al of the cases we consider $xne 0$):

Case 1: $a>0$

The function is continuous by Sqeeze theorem since $$-|x|^a<f(x)<|x|^a$$

Case 2: $a=0$

The function is continuous only if $c<0$.

Case 3: $a<0$

In this case, the function has no limit at $x=0$ when $c=0$ or $c>0$. For $c<0$ we can write:$$lim_xto 0f(x)^-c)\=lim_xto 0fracx^-a\=lim_xto 0fracx^-ccdotx\=lim_xto 0^a-c$$which is equal to zero only if $a>c$.

Conclusion

The values of $a,c$ for which $f(x)$ is continuous is as follows:$$a>0\a=0,c<0\c<a<0$$

share|cite|improve this answer

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

add a comment | 

$begingroup$

I assume the functions is $$f(x)=|x|^asin(|x|^-c)$$over $Bbb R-0$. Note that this function is even, therefore$$lim_xto 0f(x)=lim_xto 0^+f(x)$$Now, we consider $3$ cases (in al of the cases we consider $xne 0$):

Case 1: $a>0$

The function is continuous by Sqeeze theorem since $$-|x|^a<f(x)<|x|^a$$

Case 2: $a=0$

The function is continuous only if $c<0$.

Case 3: $a<0$

In this case, the function has no limit at $x=0$ when $c=0$ or $c>0$. For $c<0$ we can write:$$lim_xto 0f(x)^-c)\=lim_xto 0fracx^-a\=lim_xto 0fracx^-ccdotx\=lim_xto 0^a-c$$which is equal to zero only if $a>c$.

Conclusion

The values of $a,c$ for which $f(x)$ is continuous is as follows:$$a>0\a=0,c<0\c<a<0$$

share|cite|improve this answer

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

share|cite|improve this answer

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040