Why $a$ needs to be inside the domain of $f(x)$ in \[ \lim_{x\to a}\,f(x)=L. \]

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Lesson Summary: 
In this lesson we discuss a subtle point not needed by non-mathematics majors. Because we need to know the value of $f(x)$ in the domain $(a-\delta,a)\cup (a,a+\delta)$ to prove \[ \lim_{x\to a}\,f(x)=L, \] the number $a$ must be inside the domain and not just in it. We define the difference between in and inside.
Lesson Inputs: 
Function Limits IV: The Gory Details Of One-Sided Limits
Open and Closed Intervals
The Union and Intersection of Sets
Function Limits III: The Gory Details

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Lesson: 

Ok, this is one of those picky little details that you probably won't ever need to worry about unless you are a math major. So, if you are not a math major you can now go and feel comfortable that you don't need to understand anything in this lesson.

Now that you are still here there is one extremely minor detail that does not really add to our discussion of function limits, but does need to be addressed if we are to have a complete discussion of function limits. In our definition of function limits we said that

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means that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. What we did not emphasize at the time was that $x$ and $a$ are assumed to be in the domain of $f(x)$.

Now recall that intervals are sets and we have shown that when we say <Sign in to see all the formulas> we mean the set <Sign in to see all the formulas>.

While $a$ is assumed to be in the domain of $f(x)$ it actually needs to be inside the domain. So what do we mean by in versus inside? Let's say that we have a function whose domain is $[1,5]$. The numbers $1$ and $5$ are in $[1,5]$. By in we mean that they are elements of the set $[1,5]$. However, $1$ and $5$ are not inside $[1,5]$. The numbers $1.238$, $3.0$ and <Sign in to see all the formulas> are inside $[1,5]$, but $1$ and $5$ are not.

We require $a$ to be inside the domain so for small enough <Sign in to see all the formulas>, <Sign in to see all the formulas> and <Sign in to see all the formulas> can be in the domain. Let's keep with our example above and assume we have a function $f(x)$ whose domain is $[1,5]$. Now let's say we have some limit

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that we want to prove. The problem is that we need to know that function on <Sign in to see all the formulas>. If $a=1$ we need to know the function on <Sign in to see all the formulas>. But because the domain is $[1,5]$ we can't say anything about the function in <Sign in to see all the formulas>!

The most that we can determine would be the limit from the right:

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Since we don't even have a number for $f(x)$ in <Sign in to see all the formulas> we can't even determine a number for

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Therefore, from this problem we know that we can't determine a value for

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It's important to keep in mind that the definition of function limits is very highly restrictive. We need to be able to approach $a$ from both the left and the right, and as we approach $a$ from either side the function needs to approach the same value. If we can't do that then the most that we can say is that a left or right-sided limit exists.