# Understanding Limits - Spivak Calculus

• Sirsh
In summary, Spivak's Calculus up to chapter 5 is straightforward and easy to understand, but the concept of limits in the style/method described is difficult for some readers to grasp.
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."

Sirsh said:
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

Sirsh said:
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.

Samy_A said:
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.

Mark44 said:
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.

## What is a limit?

A limit is a fundamental concept in calculus that represents the value that a function approaches as the input approaches a certain value. It is represented by the notation lim x→a f(x) and can be thought of as the y-value of a function as the x-value gets closer and closer to a specific value on the graph.

## Why are limits important in calculus?

Limits are important in calculus because they allow us to describe and analyze the behavior of functions at specific points. They also help us understand the continuity and differentiability of functions, which are crucial concepts in calculus.

## How do you evaluate limits?

There are several methods for evaluating limits, including direct substitution, factoring, and using algebraic manipulation. Additionally, you can use L'Hopital's rule, which is a powerful tool for finding limits of indeterminate forms, or you can use graphical and numerical methods such as using a graphing calculator or creating a table of values.

## What are the types of limits?

There are three types of limits: one-sided limits, two-sided limits, and infinite limits. One-sided limits approach a specific value from one direction, while two-sided limits approach a specific value from both directions. Infinite limits occur when the y-values of a function approach positive or negative infinity as the x-values approach a certain value.

## How are limits used in real-world applications?

Limits are used in real-world applications to model and analyze various physical phenomena, such as motion, growth, and decay. For example, limits are used in physics to calculate instantaneous velocity and acceleration, and in biology to model population growth. They are also used in economics to study rates of change in supply and demand.

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