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Limits Question Definition

  • Thread starter nejnadusho
  • Start date
31
0
If I have a function lets say lim x->5 f(x) that

f(5) Does not exist, because if I give a value of x which equals the limit , there is no anymore limit?


I hope the question is understandble.

Thanks in advance.
 

Answers and Replies

Tom Mattson
Staff Emeritus
Science Advisor
Gold Member
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The existence of a limit of a function [itex]f(x)[/itex] at a point [itex]x=c[/itex] has nothing to do with whether the function is actually defined there. That's because a control is built into the definition of a limit that prevents [itex]x[/itex] from taking on the value [itex]c[/itex] as we take the limit. That's what the [itex]0<|x-c|<\delta[/itex] is there for, to ensure that the distance between [itex]x[/itex] and [itex]c[/itex] is strictly positive.
 
OrbitalPower
Yeah, I like to think of it as making sure that it is approaching c from both sides.

A good example might be [tex]\frac{\sqrt{x+1} -1}{x}[/tex]. The limit exists at c > 0 and the answer is 0 but the you can't just plug that in.

if you let x = -0.1 f(x) is 5.132 and -0.01 f(x) .5013, and -0.001 f(x) is .5001 (from the left), and x is .001, f(x) is 0.4999, x is 0.1, f(x) is 0.4988 (from the right), and so on.

When you study continuity I think it really helps clear things up in regards to the existence of limits.
 
31
0
Thank you very much
 

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