Understanding Limits - Spivak Calculus

[Sirsh]

I have read Spivak's Calculus up to chapter 5, which is on Limits. Up until this point, the majority has been very straightforward and easy to understand.

However, I am having trouble grasping the concept of limits in the style/method that Spivak describes them. Can anyone elaborate in a more general sense this definition that he has laid out, possibly with an example?

"The function f approaches the limit l near a means: for every ε > 0 there is some δ > 0 such that, for all x, if 0 < |x - a| < δ, then |f(x) - l < ε."

 

[Samy_A]

I have read Spivak's Calculus up to chapter 5, which is on Limits. Up until this point, the majority has been very straightforward and easy to understand.

However, I am having trouble grasping the concept of limits in the style/method that Spivak describes them. Can anyone elaborate in a more general sense this definition that he has laid out, possibly with an example?

"The function f approaches the limit l near a means: for every ε > 0 there is some δ > 0 such that, for all x, if 0 < |x - a| < δ, then |f(x) - l < ε."

There is a typo, it should be |f(x) - l| < ε

The first part of this Wikipedia page may help.

 

[Mark44]

I have read Spivak's Calculus up to chapter 5, which is on Limits. Up until this point, the majority has been very straightforward and easy to understand.

However, I am having trouble grasping the concept of limits in the style/method that Spivak describes them. Can anyone elaborate in a more general sense this definition that he has laid out, possibly with an example?

"The function f approaches the limit l near a means: for every ε > 0 there is some δ > 0 such that, for all x, if 0 < |x - a| < δ, then |f(x) - l < ε."

This is the usual definition of the limit of a function -- it's not specific to Spivak. Here's a very simple example: Prove that $\lim_{x \to 1} 3x = 3$.

The function here is f(x) = 3x, a straight line. It should be obvious that if x is "close to" 1, then f(x) will be "close to" 3. The limit definition quantifies all of this "close to" business.

You can think of the δ-ε business as part of a dialog between you (who are trying to prove the assertion) and an acquantance who is skeptical of the value of the limit.
Your associate provides an ε value of, say 0.06. You counter with a value of δ equal to ε/3 = 0.02. Then, if |x - 1| < 0.02. It follows from this inequality that 3|x - 1| < 3 * 0.06. In other words, that |3x - 3| < ε.

If your associate is still unconvinced, he will supply a smaller number for ε, think that that will make it harder for you. To his consternation, you come right back with a value of δ = ε/3, and show him that, indeed, if |x - 1| < δ, then |3x - 3| < ε. Eventually he will get tired of this game, and concede that $\lim_{x \to 1} 3x = 3$.

Because the function in my example is linear, it's very simple to proved the limit using the definition. Other functions require some trickery, but still use the same basic idea.

 

[Mark44]

There is a typo, it should be |f(x) - l| < ε

Good eye, Samy_A. I didn't notice that it was missing part of the absolute value.

 

[Sirsh]

This is the usual definition of the limit of a function -- it's not specific to Spivak. Here's a very simple example: Prove that $\lim_{x \to 1} 3x = 3$.

The function here is f(x) = 3x, a straight line. It should be obvious that if x is "close to" 1, then f(x) will be "close to" 3. The limit definition quantifies all of this "close to" business.

You can think of the δ-ε business as part of a dialog between you (who are trying to prove the assertion) and an acquantance who is skeptical of the value of the limit.
Your associate provides an ε value of, say 0.06. You counter with a value of δ equal to ε/3 = 0.02. Then, if |x - 1| < 0.02. It follows from this inequality that 3|x - 1| < 3 * 0.06. In other words, that |3x - 3| < ε.

If your associate is still unconvinced, he will supply a smaller number for ε, think that that will make it harder for you. To his consternation, you come right back with a value of δ = ε/3, and show him that, indeed, if |x - 1| < δ, then |3x - 3| < ε. Eventually he will get tired of this game, and concede that $\lim_{x \to 1} 3x = 3$.

Because the function in my example is linear, it's very simple to proved the limit using the definition. Other functions require some trickery, but still use the same basic idea.

Thank you very much for the explanation, makes plenty of sense to me.