Introduction to Limits

Limits: The Basic Idea

The intuitive meaning of limxcf(x)

For example: function

f(x)={x1if x!=23if x=2

The graph is:

From the graph, in this function, f(2)=2. However, limx2f(x) is not 2. It is the only value where x is very very close to 2, not actually 2. From the graph, it is easy to claim that the value where x is close to 2 is 1 (f(1.999)=0.999, f(1.99999)=0.99999, f(2.001)=1.001, f(2.00001)=1.00001). So we can write this statement as
limx2f(x)=2
And this statement also could write as f(x)1asx2.
Also, here is a important things that when you write something like limx2f(x)=2, limx2f(x) is not a function of x. We could replace x into whatever you want, like limq2f(q)=2. The letter “x” is only a temporary label for some quantity that is (in this case) getting very close to 2.

Left-Hand and Right-Hand Limits

If there is a funtion f(x), and the graph is:

极限概论

本文同时提供以下语言的翻译:English

关于极限的基本概念

The intuitive meaning of limxcf(x)

For example: function

f(x)={x1if x!=23if x=2

The graph is:

From the graph, in this function, f(2)=2. However, limx2f(x) is not 2. It is the only value where x is very very close to 2, not actually 2. From the graph, it is easy to claim that the value where x is close to 2 is 1 (f(1.999)=0.999,f(1.99999)=0.99999,f(2.001)=1.001,f(2.00001)=1.00001). So we can write this statement as
limx2f(x)=2
And this statement also could write as f(x)1asx2.
Also, here is a important things that when you write something like limx2f(x)=2, limx2f(x) is not a function of x. We could replace x into whatever you want, like limq2f(q)=2. The letter “x” is only a temporary label for some quantity that is (in this case) getting very close to 2.

Left-Hand and Right-Hand Limits

If there is a funtion f(x), and the graph is:

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