Having trouble with limits and continuity? Let's clear things up!

In summary, the conversation discusses limits involving 0+ and 0-, and how to obtain the answers for these limits. It also touches on the concept of continuity and the definition of Cn. The conversation also delves into the definition of f(0) and how it can be proven that f(0) = 1 using L'Hopital's rule. Finally, the conversation concludes with a discussion on the definition of f(0) and how it is a reasonable choice for the function to be continuous at that point.
  • #1
daster
I'm having trouble with limits that involve 0+ and 0-. Can someone show me how the answers to the following limits are obtained?

[tex]f(x) = \frac{1}{1+e^{\frac{1}{x}}}[/tex]

[tex]\lim_{x\rightarrow0^{+}} = 0[/tex]

[tex]\lim_{x\rightarrow0^{-}} = 1[/tex]

Now, my second query involves continuity. I understand that:

[tex]f(x) \in C \Leftrightarrow \lim_{x \rightarrow a} f(x) = f(a)[/tex]

Say we have:

[tex]f(x) = \frac{\sin x}{x}[/tex]

Is f(x) continuous at x=0? My book says it is if f(0) is defined as 1. What am I missing?

Finally, what exactly is Cn?
 
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  • #2
as x tends to zero and is positive, 1/x tends to infinity so e^{1/x} tends to infinity, hence f tends to zero as x tends to 0 from above

as x tends to zero from below, 1/x goes it minus infinity, and e^[1/x} goes to zero hence f tends to 1 as x goes to 0 from below.

f is continuous at 0 with the assignment of f(0)=1 this can be proven by, say, l'hopital's rule.

C^n is the space of functions that are differentiable n times and where the n'th derivative is continuous. C is the continuous functions.

eg as functions on R |x| is C but not C^1, x|x| is C^1 but not C^2
 
  • #3
Thanks for the reply matt. I just have two questions:

e^(1/x) goes to zero if 1/x goes to minus infinity?

How is f(0)=1?
 
  • #4
1+e^{1/x} tends to 1, doesn't it? so 1/(1+e^{1/x}) tends to 1 as wel. all as x tends to 0 from below
 
  • #5
Well yes, but I'm not understanding how e^(1/x) tends to zero if (1/x) tends to minus infinity. Shouldn't it also tend to minus infinity?
 
  • #6
you do know what the graph of e^x looks like?

e^-x = 1/e^x, so if e^x goes to INFINITY (CORRECTED TYPO) as x goes to infinty then e^x must tend to zero as x goes to minus infinity.
 
Last edited:
  • #7
But you said in your first post:
"1/x tends to infinity so e^{1/x} tends to infinity"
 
  • #8
and that is correct whilst x is always positive. so?


are you referring to my typo that i'll correct
 
  • #9
If you look at the graph, when x tends to -oo then e^x tends to zero

set x=1/u and you have

(1/u)->-oo => e^(1/u)->0
 
  • #10
Oh! So if x tends to minus infinity, e^x is actually e^(-x) where x tends to infinity and thus e^x tends to 1/infinity=0? I think I got it now. Thanks matt. :smile:

I know I'm being a bother but... My second question was how can f(0)=1 (where f(x)=sinx/x)?
 
  • #11
like i said, l'hopital's rule, though this is sort of a cheat.

there is a geometric proof somewhere, but i don't konw where to find a copy of it.
 
  • #12
So f(0)=1 is actually lim[f(x)] as x tends to 0?
 
  • #13
the limit of f(x) is 1 when (x->0+) or (x->0-)

By the way f(x) can't have a value when x=0

But if we define - as you said - that f(0)=1,
then we made f(x) continuous
(left limit = right limit = f(0))
 
  • #14
So f(0)=1 is just a definition? But does that mean that it's not necessarily true?
 
  • #15
I'll recall another section

what is a power of a^0 ?

By definition, a^n=a*a*...a (n factors)
we can't find out what a^0 means

but a^0=1 works (if, for example, we think of (a^7)/(a^7)=1)

So we DEFINE a^0=1

The same as 0! (factorial) - we define it although there is not a factorial

If you read at your book that we define f(0)=1, that is,
we do that just because it works!

Actually there is not f(0) (sin0/0)
It's just a definition , but this still works
 
  • #16
if we chose any other number (and we must choose some number to make it a function from R to R) then it wouldn't be continuous there, so it is a reasonalbe choice - remember you must define the function in some way for all points in the domain or it isn't a function.
 
  • #17
Okay. So f(0)=1 is simply a defined value.

Thanks for your help guys.
 
  • #18
yes, it's called a removably singularity: although the function (sinx)/x isn't "techincally" defined at 0 it is clear how he ought to define it
 

1. What are limits and continuity?

Limits and continuity are concepts in calculus that deal with the behavior of a function as it approaches a certain value or point. Limits determine the value that a function approaches as its input approaches a specific value, while continuity refers to the smoothness and connectedness of a function over its entire domain.

2. Why is understanding limits and continuity important?

Understanding limits and continuity is important because they form the foundation of calculus, which is a fundamental branch of mathematics used in various fields such as physics, engineering, and economics. These concepts are also crucial in solving more complex problems in mathematics and real-life applications.

3. What are some common mistakes when dealing with limits?

Some common mistakes when dealing with limits include not considering the behavior of the function as it approaches the limit, not simplifying the expression before evaluating the limit, and not using the correct rules and theorems for evaluating limits. It is important to carefully follow the steps and rules when working with limits to avoid these mistakes.

4. How can I improve my understanding of limits and continuity?

Improving your understanding of limits and continuity requires regular practice and a strong foundation in algebra and trigonometry. It is also helpful to work through various examples and problems, seek help from a tutor or teacher, and use online resources such as videos and practice exercises to reinforce your understanding.

5. Are there any real-life applications of limits and continuity?

Yes, there are many real-life applications of limits and continuity. For example, they are used in physics to analyze the motion of objects, in economics to model and predict market trends, and in engineering to design structures and systems. These concepts also have applications in computer science, biology, and other fields.

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