Finding Limits Using Properties - Simplifying Examples

Anybody who cannot cancel a factor of x - 1 from (x - 1)(x + 10) has no business doing Calculus. If you don't get the fundamentals down, you will never learn Calculus.In summary, the conversation discussed using the properties of limits to find the limit of a given function. The final answer was 18, but there was some disagreement and discussion about the approach and attitude towards learning math. The importance of reviewing Intermediate Algebra and understanding fundamental concepts was emphasized.
  • #1
nycmathguy
Homework Statement
Use properties of limits to find the limit.
Relevant Equations
N/A
Use the properties of limits to find the limit.

lim [x(x − 1)(x + 10)]
x→−1

lim [x(x^2 + 9x - 10)]
x→−1

lim [x^3 + 9x^2 - 10x]
x→−1

lim (x^3) + lim (9x^2) - (10x)
x→−1 x→−1 x→−1.

(-1)^3 + 9(-1)^2 - 10(-1)

-1 + 9 -(-10)

-1 + 9 + 10

-1 + 19 = 18

The limit is 18.

You say?
 
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  • #2
Yes.

Notice that these are all continuous functions. Given this, you could plug -1 in at the beginning giving -1*-2*9=18
 
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  • #3
caz said:
Yes.

Notice that these are all continuous functions. Given this, you could plug -1 in at the beginning giving -1*-2*9=18
Yes, but the instructions are to use the properties of limits in section 1.2 of chapter 1. I could have just plugged in (-1) for x and call it a day. This is an exercise in terms of using the properties of limits.
 
  • #4
Ok
 
  • #5
caz said:
Ok
With FB math groups becoming more popular as the years go by, the creator of this site should appreciate all my threads without picking on me for insignificant matters. Do you agree?
 
  • #6
nycmathguy said:
the creator of this site should appreciate all my threads without picking on me for insignificant matters.
To you they are insignificant, but many of these "insignificant" matters contribute to your difficulty in solving the problems you post.

nycmathguy said:
Do you agree?
No, I don't.
 
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  • #7
I am not trying to be harsh, but no.
1) We’re volunteers. You should be appreciating us, not the other way around. Your biggest worry should be burning people out so that they will not even look at your questions.
2) Most people do not adequately search, so the idea that you are creating valuable archival content is mistaken.

My advice is to only ask questions that concern you. I will grant you that some of us are abrasive, but do not argue too much. You are here because we have learned things that you haven’t yet.
 
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  • #8
caz said:
I am not trying to be harsh, but no.
1) We’re volunteers. You should be appreciating us, not the other way around. Your biggest worry should be burning people out so that they will not even look at your questions.
2) Most people do not adequately search, so the idea that you are creating valuable archival content is mistaken.

My advice is to only ask questions that concern you. I will grant you that some of us are abrasive, but do not argue too much. You are here because we have learned things that you haven’t yet.
Ok. I will take a break for today and return tomorrow to continue discussion for questions posted today. No need to post new questions daily.
 
  • #9
You want to be responsive on open threads. You should think about how many threads you open.
What you want to do is keep progressing. Our reward is watching your understanding grow.
 
  • #10
nycmathguy said:
Ok. I will take a break for today and return tomorrow
And then we have this. So much for promises.

You seem to think we need you more than you need us. Wrong. People are are very willing to help, but not if they are treated like something unpleasant stuck to the bottom of their shoes.

You have a few things standing in the way of progress. One is the great big chip - almost a boulder - on your shoulder. You are rapidly reaching a point where you have to choose between that chip and further understanding of math.

Another is that you think you know better what is useful and important than those who have already gotten to where you want to be. That is very foolish. If you don't drop that attitude, you aren't going to get anywhere.

Finally, berating @caz for giving an answer that doesn't use a list of properties in "section 1.2 of chapter `" of some unknown book before you even mentioned this is...um...a good example of how not to learn.
 
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  • #11
Vanadium 50 said:
And then we have this. So much for promises.

You seem to think we need you more than you need us. Wrong. People are are very willing to help, but not if they are treated like something unpleasant stuck to the bottom of their shoes.

You have a few things standing in the way of progress. One is the great big chip - almost a boulder - on your shoulder. You are rapidly reaching a point where you have to choose between that chip and further understanding of math.

Another is that you think you know better what is useful and important than those who have already gotten to where you want to be. That is very foolish. If you don't drop that attitude, you aren't going to get anywhere.

Finally, berating @caz for giving an answer that doesn't use a list of properties in "section 1.2 of chapter `" of some unknown book before you even mentioned this is...um...a good example of how not to learn.
1. You are wrong.

2. I got bored and decided to post a few more questions. Big deal.

3. You say I am proud without knowing me personally.

4. You are more than welcome to skip all my threads. Why torture yourself?

5. I appreciate what caz did for me but going through the textbook one chapter, one section at a time means exactly that. It's so much easier to evaluate limits but section 1.2 tells me to use properties of limits.

6. You don't know Ron Larson? Some unknown book? How many times must I repeat myself? PRECALCULUS BY RON LARSON. EDITION 10E. Look it up.

7. Read 4 above again. How about two more times?
 
  • #12
nycmathguy said:
Homework Statement:: Use properties of limits to find the limit.
Relevant Equations:: N/A

Use the properties of limits to find the limit.

lim [x(x − 1)(x + 10)]
x→−1

lim [x(x^2 + 9x - 10)]
x→−1

lim [x^3 + 9x^2 - 10x]
x→−1

lim (x^3) + lim (9x^2) - (10x)
x→−1 x→−1 x→−1.

(-1)^3 + 9(-1)^2 - 10(-1)

-1 + 9 -(-10)

-1 + 9 + 10

-1 + 19 = 18

The limit is 18.

You say?
You need to review Intermediate Algebra intensively, all of it; Once that is done, you can restudy Pre-Calculus slowly, intensively, and then you may find in your book, some introduction sections on Limits. All of this including studying Limits in the book's presented manner, will be time consuming and you will need to check the book instruction, the example exercises, and the section exercises in careful detail. You must be willing to repeat what you study in these sections and the included exercises, on your own.

Studying Limits from a Pre-Calculus textbook requires long, daily effort.
 
  • #13
nycmathguy said:
With FB math groups becoming more popular as the years go by, the creator of this site should appreciate all my threads without picking on me for insignificant matters. Do you agree?
No. And you are picking that the PF creator is picking on you, not sure how.
 
  • #14
nycmathguy said:
Yes, but the instructions are to use the properties of limits in section 1.2 of chapter 1. I could have just plugged in (-1) for x and call it a day. This is an exercise in terms of using the properties of limits.
I agree with @caz. Ultimately, you are plugging in the value ##x = -1## here. Whether you split the expression up or not makes no difference. If I plug ##x = -1## into the initial expression, I'll get the same answer as you.

Also, since there is nothing problematic about this function, this has nothing to do with limits, as such. You are simply being asked to calculate a function value for a given value of ##x##. Which is what you've done.
 
  • #15
I think the OP is not introduced yet (at least through his textbook) to the concept of continuous or discontinuous functions.
 
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  • #16
Delta2 said:
I think the OP is not introduced yet (at least through his textbook) to the concept of continuous or discontinuous functions.
A maths student should be asking how it's possible that ##\lim_{x \rightarrow a}f(x) \ne f(a)##? In this case we have that ##f(x)## is a polynomial. Why can't we simply calaculate ##f(-1)##? How can we possibly get a different answer by splitting the polynomial up? That would be mathematical thinking. And, by asking that question, the student might learn something about mathematics.

The alternative, to blindly follow some prescribed steps could be seen as un-mathematical. That might lead to the situation where the student can only do precisely what they have been taught and has no facility to check or analyse their work themselves. And, indeed, never develops the facility to think mathematically.

The process becomes:

1) Memorise some steps.

2) Carry out those steps for a given problem.

3) Ask someone who knows whether the answer is correct.

That seems to me a tragedy for school children or an adult student who wants to learn mathematics.
 
  • #17
posts #1 and #16:
He has a polynomial but as its factored form. He mentioned wanting to use properties of limits and these seem to have been missed/overlooked. BUT he then seemed to have used them maybe or maybe not recognizing.

Most of the steps look good, but not this particular step:
lim (x^3) + lim (9x^2) - (10x)
x→−1 x→−1 x→−1.
 
Last edited:
  • #18
PeroK said:
A maths student should be asking how it's possible that ##\lim_{x \rightarrow a}f(x) \ne f(a)##? In this case we have that ##f(x)## is a polynomial. Why can't we simply calaculate ##f(-1)##? How can we possibly get a different answer by splitting the polynomial up? That would be mathematical thinking. And, by asking that question, the student might learn something about mathematics.

The alternative, to blindly follow some prescribed steps could be seen as un-mathematical. That might lead to the situation where the student can only do precisely what they have been taught and has no facility to check or analyse their work themselves. And, indeed, never develops the facility to think mathematically.

The process becomes:

1) Memorise some steps.

2) Carry out those steps for a given problem.

3) Ask someone who knows whether the answer is correct.

That seems to me a tragedy for school children or an adult student who wants to learn mathematics.
I think you want to take him out of the "bubble" of his textbook too early. For the time being I would recommend that he follows his textbook, so for this problem he just is interested to apply properties of limits like that ##\lim(f(x)+g(x))=\lim f(x)+\lim g(x)## or ##\lim\lambda f(x)=\lambda\lim f(x)##. Questions about the discontinuity of functions could come to him later when his textbook introduces him to that concept.
 
  • #19
Delta2 said:
I think you want to take him out of the "bubble" of his textbook too early. For the time being I would recommend that he follows his textbook, so for this problem he just is interested to apply properties of limits like that ##\lim(f(x)+g(x))=\lim f(x)+\lim g(x)## or ##\lim\lambda f(x)=\lambda\lim f(x)##. Questions about the discontinuity of functions could come to him later when his textbook introduces him to that concept.
That leaves the problem of how one calculates any limit. In your example, how do you calculate the limit of ##f(x)##? You plug in ##-1##. But, you can't just plug in ##-1## to ##f(x) + g(x)##? Why? Because the book says so!

But, then, as we see the student must post every problem online with the question "is this correct"?

How is that learning mathematics?
 
  • #20
PeroK said:
That leaves the problem of how one calculates any limit. In your example, how do you calculate the limit of f(x)? You plug in −1. But, you can't just plug in −1 to f(x)+g(x)? Why? Because the book says so!
Because the book says to use properties of limit.
I think you are a strongly sparkling spirit Perok (direct translation from Greek, not sure how the phrase in English is). The OP might not be so strongly sparkling :D.

PeroK said:
But, then, as we see the student must post every problem online with the question "is this correct"?

How is that learning mathematics?
Well yes , the OP posts many similar problems when only 1-2 of them would be enough. Either he doesn't understand the similarity or he wants to build confidence.
 
  • #21
I did precalculus in 9th grade. If I remember correctly, roughly 36 weeks of daily instruction with multiple graded problem sets each week. I had classmates that I could also ask questions to and compare answers with. Very interactive.

nycmathguy hasn’t taken a class for decades and is doing this by himself. I think we should cut him some slack. He is still learning how to learn.
 
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  • #22
Delta2 said:
I think the OP is not introduced yet (at least through his textbook) to the concept of continuous or discontinuous functions.
Bingo!
 
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  • #23
caz said:
I did precalculus in 9th grade. If I remember correctly, roughly 36 weeks of daily instruction with multiple graded problem sets each week. I had classmates that I could also ask questions to and compare answers with. Very interactive.

nycmathguy hasn’t taken a class for decades and is doing this by himself. I think we should cut him some slack. He is still learning how to learn.
Thank you. I took precalculus in the Spring 1993 semester at Lehman College.
 

1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as the input values approach a certain value. It is the value that a function approaches, but may not necessarily reach, as the input values get closer and closer to a specific value.

2. How do you find a limit using properties?

To find a limit using properties, you can use algebraic manipulations and known properties of limits, such as the sum, difference, product, quotient, and power properties. These properties allow you to simplify the function and evaluate the limit at a specific value.

3. Can you find a limit using properties for all functions?

No, not all functions can be evaluated using properties. Some functions may have more complex behaviors that cannot be simplified using known properties. In these cases, other methods, such as graphing or using L'Hospital's rule, may be necessary to find the limit.

4. What are some common mistakes when finding limits using properties?

One common mistake is applying the properties incorrectly, especially when dealing with more complex functions. It is important to carefully follow the rules and make sure all steps are mathematically valid. Another mistake is forgetting to check for any restrictions on the function, which may affect the limit value.

5. How can finding limits using properties be useful in real-world applications?

Finding limits using properties is useful in many real-world applications, such as in physics, economics, and engineering. It allows us to understand the behavior of a system or function as it approaches a certain value, which can help in making predictions and decisions. For example, in economics, limits can be used to analyze the maximum profit a company can make or the maximum production capacity of a factory.

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