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Lean Calculus

Dr. Chad Scott

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Section 1.1 The Main Idea

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Consider the function
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sin(x)/x
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Notice that this function can’t be evaluated at x=0 (i.e. 0 is not in its domain.) But, we could compute it for values of x *close* to 0. For example, consider the following table for sin(x)/x.
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x sin(x)/x
0.500 0.958851
0.100 0.998334
0.010 0.999983
0.001 0.999999
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Look at this table. As we move the x values closer to 0, what are the sin(x)/x values getting closer to? It’s apparent that as x approaches 0, that sin(x)/x approaches 1. Of course, to be fair, at the moment we’re only letting x approach 0 from the right hand (positive) side of 0. Let’s create some notation to record this kind of information.
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Definition 1.1.1.

If f(x) approaches a number, say l, as x approaches a number a from its right hand side, then we write
limx→a+f(x)=l
which is pronounced "The limit of f(x) is l as x approaches a from the right". We also say that f(x) has a right hand limit of l at a. Analogously, If f(x) approaches a number, say k as x approaches a from its left hand side, then we write
limx→a−f(x)=k
which is pronounced "The limit of f(x) is k as x approaches a from the left". Again, we also say that f(x) has a left hand limit of k at a.
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So now, with this new notation, it seems reasonable to write
limx→0+(sin(x)/x)=1
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Remark 1.1.2.

It is critical to note here that we are using the word "approach" in the definition above to indicate that we simply take values of the domain variable x *close* to a but not equal to it. In computing limits, we do not use x=a. We’re just identifying what f(x) is becoming as x *approaches* a. In fact, in many cases such as sin(x)/x, we can’t even compute the function at x=a=0.

Example 1.1.3.

For f(x)=sin(x)/x from above, make a table like the one above but this time use 4 or 5 domain values approaching 0 from the left. Use the notation from the definition above to write your conclusion about the left hand limit.
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Finally, since it will often be the case that both the left hand and right hand limit come out the same (but not always) for a function, we just drop the + and − in that case. In other words ...
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Definition 1.1.4.

In case both the left hand and right hand limit of a function are the same (i.e. limx→a+f(x)=limx→a−f(x)=l), then we simply write limx→af(x)=l without any reference to + or −. In this case we say that f(x) has a limit of l at a.

Example 1.1.5.

For f(x)=|x|/x, notice that once again 0 is not in its domain. Make tables for left and right hand approaches to 0 and use proper notation as defined above to write your conclusions about right hand limit, left hand limit and limit.
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