Prove Limit Existence: sin(1/x) as x→0

  • Context: Graduate 
  • Thread starter Thread starter aquitaine
  • Start date Start date
  • Tags Tags
    Existence Limits
Click For Summary
SUMMARY

The limit of sin(1/x) as x approaches zero does not exist due to the discontinuity of the function 1/x at x=0. The left-sided limit approaches negative infinity while the right-sided limit approaches positive infinity, violating the definition of a limit. According to the formal definition, for every ε>0, there must exist a δ>0 such that if |x-x0|<δ, then |f(x)-L|<ε. Since this condition fails at x0=0, the limit is confirmed to be nonexistent.

PREREQUISITES
  • Understanding of limit definitions in calculus
  • Knowledge of discontinuous functions
  • Familiarity with left-sided and right-sided limits
  • Basic trigonometric functions, specifically sine
NEXT STEPS
  • Study the formal definition of limits in calculus
  • Explore discontinuities in functions and their implications
  • Learn about left-hand and right-hand limits in detail
  • Investigate the behavior of trigonometric functions near discontinuities
USEFUL FOR

Students of calculus, mathematics educators, and anyone interested in understanding limit behavior in functions, particularly those involving trigonometric expressions.

aquitaine
Messages
30
Reaction score
9
In general, what is a good method to prove whether or not a limit exists? For example limit of sin(1/x) as x approaches zero.
 
Physics news on Phys.org
Hi, ill recommend you to take a look to the definition of limit:
For every 'e'>0,exists 'd'>0, such that for /x-x0/<d then
/f(x)-L/<e
If that holds, then the function f(x) has limit at the point x0,
and it is equal to L.
Of course a bold statement like this doesn't seem to help to much, so here is what you do for practical purpouses:
the function 1/x has a discontinuity at x=0, so evaluate the
rightsided and leftsided limits at x=0:
lim x->0-...1/x=-inf, and lim x->0+...1/x=+inf.
Then the definition of limit doesn't hold for x0=0 and therefor the limit doesn exists at this point.
Hope i helped you.
Have a good day.


/
 

Similar threads

Undergrad Multivariable limits - path problem
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Vector calculus - Prove a function is not differentiable at (0,0)
  • · Replies 1 ·
Replies
1
Views
2K
High School Wondering if a limit exists or not
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad What's my mistake in this integration problem?
  • · Replies 8 ·
Replies
8
Views
3K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Limit as a function, not a value
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Showing that a multivariable limit does not exist
  • · Replies 3 ·
Replies
3
Views
5K
MHB How to Use the Squeeze Theorem to Find This Limit?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Proving Limit of Piecewise Function at x=0
  • · Replies 2 ·
Replies
2
Views
3K
MHB Is the Limit of Sin x/x=1 Proven in Elementary Calculus?
  • · Replies 3 ·
Replies
3
Views
2K