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Prove discontinuity at given point

-1

$begingroup$

Given the function $y= lim_{n to infty} frac{1}{1+x^n}$ for $x geq 0$, show that the function is discontinuous at $x=1$?

I tried the question , it comes out to be continuous using left hand limit and right hand limit rule

calculus limits continuity discontinuous-functions

share|cite|improve this question

edited Mar 23 at 12:32

Yanior Weg

2,85611547

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  If $xin(0,1)$ then $x^nto0$ as $ntoinfty$. Therefore, $frac{1}{1+x^n}tofrac{1}{1+0}=1$. However, if $x=1$ then $frac{1}{1+x^n}=frac{1}{1+1}=frac{1}{2}$. Therefore, the limit function has limit from the left at $x=1$ that is equal to $1$, but the value at the point is $1/2$.
  $endgroup$
  – user647486
  Mar 23 at 12:19
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Is this question about one function or a sequence of functions? Is it about continuity or about a limit?
  $endgroup$
  – Kavi Rama Murthy
  Mar 23 at 12:20
  

  
  
  
  

add a comment | 

-1

$begingroup$

Given the function $y= lim_{n to infty} frac{1}{1+x^n}$ for $x geq 0$, show that the function is discontinuous at $x=1$?

I tried the question , it comes out to be continuous using left hand limit and right hand limit rule

calculus limits continuity discontinuous-functions

share|cite|improve this question

edited Mar 23 at 12:32

Yanior Weg

2,85611547

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  If $xin(0,1)$ then $x^nto0$ as $ntoinfty$. Therefore, $frac{1}{1+x^n}tofrac{1}{1+0}=1$. However, if $x=1$ then $frac{1}{1+x^n}=frac{1}{1+1}=frac{1}{2}$. Therefore, the limit function has limit from the left at $x=1$ that is equal to $1$, but the value at the point is $1/2$.
  $endgroup$
  – user647486
  Mar 23 at 12:19
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Is this question about one function or a sequence of functions? Is it about continuity or about a limit?
  $endgroup$
  – Kavi Rama Murthy
  Mar 23 at 12:20
  

  
  
  
  

add a comment | 

$begingroup$

Given the function $y= lim_{n to infty} frac{1}{1+x^n}$ for $x geq 0$, show that the function is discontinuous at $x=1$?

I tried the question , it comes out to be continuous using left hand limit and right hand limit rule

calculus limits continuity discontinuous-functions

share|cite|improve this question

edited Mar 23 at 12:32

Yanior Weg

2,85611547

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43

$endgroup$

share|cite|improve this question

edited Mar 23 at 12:32

Yanior Weg

2,85611547

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43

share|cite|improve this question

edited Mar 23 at 12:32

Yanior Weg

2,85611547

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43

edited Mar 23 at 12:32

Yanior Weg

2,85611547

edited Mar 23 at 12:32

Yanior Weg

2,85611547

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43

asked Mar 23 at 12:14

Sanjam MannSanjam Mann

43