What Are the Properties of Limits in Mathematics?

I think I know what you're talking about. It's a limit without a limit. In summary, the conversation discusses taking the cube root of 8 and then cubing it, and the resulting limit being 1/2. The conversation also touches on continuity and discontinuity, and the presence of limits in a precalculus book.
  • #1
nycmathguy
Homework Statement
Determine the limit of f(x) using properties of limits.
Relevant Equations
N/A
See attachment for question and math work.
 

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  • #2
You took the cube root of 8 (81/3) and then cubed it. Why?
 
  • #3
Doc Al said:
You took the cube root of 8 (81/3) and then cubed it. Why?
Let's take it from here:

(1/4)[cr{8}]

(1/4)(2)

1/2

The limit is 1/2.

Why cubed the cube root of 8 in my first attempt?

Answer: typo
 
  • #4
OK, now you've got it.
 
  • #5
Yes the limit is 1/2. If we tweak the function and we make it so that f(x) is the same as before for all ##x\neq 8## but we set ##f(8)=256## what will the limit be?
 
  • #6
Delta2 said:
Yes the limit is 1/2. If we tweak the function and we make it so that f(x) is the same as before for all ##x\neq 8## but we set ##f(8)=256## what will the limit be?
The limit would be 256.
 
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  • #7
nycmathguy said:
The limit would be 256.
Nope it will remain 1/2. For the limit we focus what happens around the point of interest but not necessarily onto the point of interest. Around 8 the tweaked function remains the same so the limit remains the same. I just introduced an artificial discontinuity in the tweaked function by setting f(8)=whatever except 1/2.

I guess you need to be introduced to continuity and discontinuity by your textbook. If it is not done by your precalculus book, it should be done by your calculus I book.
 
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  • #8
Delta2 said:
Nope it will remain 1/2. For the limit we focus what happens around the point of interest but not necessarily onto the point of interest. Around 8 the tweaked function remains the same so the limit remains the same. I just introduced an artificial discontinuity in the tweaked function by setting f(8)=whatever except 1/2.

I guess you need to be introduced to continuity and discontinuity by your textbook. If it is not done by your precalculus book, it should be done by your calculus I book.
What are limits doing in a pre-calculus book, one wonders?
 
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  • #9
PeroK said:
What are limits doing in a pre-calculus book, one wonders?
The boundaries between calculus and precalculus are fuzzy, at least that's what Ron Larson thinks lol...
 
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  • #10
PeroK said:
What are limits doing in a pre-calculus book, one wonders?
Not in a precalculus book. I made a typo. The limits are in the following book:
 

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What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It represents the value that the function approaches, or "approaches but never quite reaches", as the input gets closer and closer to the specified value.

How do you find the limit of a function?

The limit of a function can be found by evaluating the function at values close to the specified limit and observing the trend of the outputs. If the outputs approach a certain value as the inputs get closer to the specified value, then that value is the limit of the function.

What are some properties of limits?

Some properties of limits include the sum, difference, product, and quotient properties, which state that the limit of a sum, difference, product, or quotient of two functions is equal to the sum, difference, product, or quotient of the limits of the individual functions.

What is the Squeeze Theorem and how is it used?

The Squeeze Theorem, also known as the Sandwich Theorem, states that if two functions have the same limit at a certain point, and a third function is between them, then the third function also has the same limit at that point. It is used to evaluate limits of functions that are difficult to evaluate directly.

Can a function have a limit at a point but not be continuous at that point?

Yes, a function can have a limit at a point but not be continuous at that point. This can happen if there is a "hole" or "jump" in the graph of the function at that point, or if the function is undefined at that point. In order for a function to be continuous at a point, it must have a limit at that point and the limit must be equal to the value of the function at that point.

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