Limits problem?

No matter what x equals, as long as x is not 0, x/x is the constant 1.

The limit you posted means that x is never 0. x may get very close to 0, but x will never equal 0.

What if x/x was replaced with 1, what would you think the limit equals?
 
The statement that [imath]\mathop {\lim }\limits_{x \to A} (x) = B[/imath]
means that for each [imath]x\sim A[/imath] , [imath] x\ne A[/imath] then [imath]f(x)\sim B[/imath]
That reads: if x is close to A but not A then f(x) is close to B.
Now please answer your own question.
 
You can always do this while evaluating limits

\(\displaystyle \lim_{x \to 0} \frac{x}{x} = \lim_{x \to 0} 1 = 1, \ x \neq 0 \).


A similar example is

\(\displaystyle \lim_{x \to 1} \frac{x - 1}{x - 1} = \lim_{x \to 1} 1 = 1, \ x \neq 1 \).
 
No it is not a special rule. There is a ton of mystification about limits.

INFORMALLY, if f(x) = x/x, f(x) does not exist if x = 0. Define g(x) = 1. That is defined everywhere. Moreover g(x) = f(x) for all x if not x not equal to 0.

Now apply the formal definition of limit.
 
Yes, it is a special limit, then again all limits are special.

x/x = 1, if x is not 0. So you can replace x/x with 1.

\(\displaystyle \lim_{x \to 0} 1 = 1 \)
A constant never varies even as x changes.[/tex]
 
EddyBenzen122 said:
So the limit is equal to one because as long as x is not zero it’d always equal to one and so we can say that if x is zero it wouldn’t equal to one, instead it approaches one.
Am I correct?
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There are two formal definitions of limit, one due to Weierstrass (standard) and one due to Robinson (non-standard), I am not familiar with Robinson’s approach.

Here is the more standard approach

[math]\text {If for ANY positive number } \epsilon, \exist \ \delta \text { such that}[/math]
[math]\delta \ne 0 \text { and } |x - c| < \delta \implies |L - f(x)| < \epsilon,[/math]
[math]\text {then the limit of } f(x) \text { as } x \text { approaches } c \text { is } L.[/math]
If you think about what that means, you will see that it means that if x is not far from c, f(x) is not far from L. The formal presentation is very abstract and took decades to develop. But the intuition is simple, the closer x gets to c, the closer f(x) gets to L, the limit.

Let’s apply that to this example.

[math]|x \ne 0 \implies |f(x) - 1| = |1 - 1| = |0| = 0 < \text { any positive number.}[/math]
So 1 is the limit. This is a very simple example. They can get mind bendingly complex, but the intuition is simple.
 
JeffM said:
There are two formal definitions of limit, one due to Weierstrass (standard) and one due to Robinson (non-standard), I am not familiar with Robinson’s approach.

Here is the more standard approach

[math]\text {If for ANY positive number } \epsilon, \exist \ \delta \text { such that}[/math]
[math]\delta \ne 0 \text { and } |x - c| < \delta \implies |L - f(x)| < \epsilon,[/math]
[math]\text {then the limit of } f(x) \text { as } x \text { approaches } c \text { is } L.[/math]
If you think about what that means, you will see that it means that if x is not far from c, f(x) is not far from L. The formal presentation is very abstract and took decades to develop. But the intuition is simple, the closer x gets to c, the closer f(x) gets to L, the limit.

Let’s apply that to this example.

[math]|x \ne 0 \implies |f(x) - 1| = |1 - 1| = |0| = 0 < \text { any positive number.}[/math]
Click to expand...
you put "1" for c, does that mean c is always "1"? Where did c = 1 come from?
 
Jomo said:
No, sorry but what you're saying is not true. In one place you put it for the limit of f(x) and in another place you put it for f(x).
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if i had said when x aproaches 0, the limit of f(x) when x approaches 0 is "1", would that be correct?
 
EddyBenzen122 said:
if i had said when x aproaches 0, the limit of f(x) when x approaches 0 is "1", would that be correct?
Click to expand...
Yes, a bit redundant, but you have it.

I might put it into English like this

As x approaches 0, f(x) = x/x approaches or equals 1, which is the limit of f(x) as x approaches 0.

It is important to realize that, in standard analysis, x must not equal what it is approaching, but f(x) may equal the limit.

It is subtle definition and takes a while to get comfortable with.
 
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