Why does the quotient law not work here?

The quotient rule can't be used because the quotient rule is to find the derivative of sinx/x. Why would you want the derivative of sinx/x if you are asked to find a limit.

If you are thinking of L'Hopital rule, the rule does NOT say to compute the derivative of the fraction. Rather it says to compute the derivative of the numerator and denominator separately and then divide. Now take the lim of that fraction.

Yes, x gets very close to 0 but NEVER equals 0. One way to compute the limit is to let x=0.00001 and compute sin0.00001/0.00001 and you will get a result very close to 1. If you do not believe it is 1 (maybe you believe the limit is 1.00000001), then pick an x value even closer to 0.
 
Integrate said:
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Doesn't a limit never actually meet zero? Only infinitely close?
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Look at the statement of the Quotient Law (for limits, I presume!) as given to you; if you don't see the answer, quote it to us and tell us what you think specifically about it.

I wonder if your question is about something slightly different; it reminds me of questions people often ask about asymptotes (which are another kind of limit). Are you wondering about the fact that this function equals zero infinitely many times?
 
lev888 said:
The quotient rule can't be used if the limit of the denominator is 0.
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lev, isn't it true that you can't use the quotient law for limits only if the numerator and denominator are both 0?
 
Jomo said:
lev, isn't it true that you can't use the quotient law for limits only if the numerator and denominator are both 0?
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You may be right. I studied this 30 years ago. Couple of sites I looked up before posting mentioned only the denominator.
 
lev888 said:
You may be right. I studied this 30 years ago. Couple of sites I looked up before posting mentioned only the denominator.
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If after direct substitution you get 7/0, you are done. I think I am right but I took calculus 1 back in 1986 (just over 30 years ago) and haven't taught it (or anything) for 11 years.
 
Dr.Peterson said:
Look at the statement of the Quotient Law (for limits, I presume!) as given to you; if you don't see the answer, quote it to us and tell us what you think specifically about it.

I wonder if your question is about something slightly different; it reminds me of questions people often ask about asymptotes (which are another kind of limit). Are you wondering about the fact that this function equals zero infinitely many times?
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Quotient Law.png

As long as the function in the denominator does not equal zero the quotient law works... hmm this gives me a few more conflicts.

The original question doesn't have a function in the denominator. Only a variable. Does this have an affect?

And why would the law not allow for the function to approach zero if the denominator never actually is zero?
 
OK, OK, it is late here. The above rule is true. Actually my rule, which has no name, is also true. How about we call this a draw.
I really hate some definitions as recalling them is meaningless to the understanding of mathematics.
 
Integrate said:
View attachment 24473

As long as the function in the denominator does not equal zero the quotient law works... hmm this gives me a few more conflicts.

The original question doesn't have a function in the denominator. Only a variable. Does this have an affect?

And why would the law not allow for the function to approach zero if the denominator never actually is zero?
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Isn't g(x) = x a function? It's not a matter of how it's written, but of what it is! And any expression is a function.

Limits can exist when the denominator approaches zero, but you can't use this rule to find them. At the end you'd have to divide by that limit of zero, and you can't do that!
 
Dr.Peterson said:
Isn't g(x) = x a function? It's not a matter of how it's written, but of what it is! And any expression is a function.

Limits can exist when the denominator approaches zero, but you can't use this rule to find them. The proof of the theorem would show you why.
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I am not clear what you are saying. Suppose we have lim⁡x−>5xx−5\lim_{x->5}\dfrac{x}{x-5}. I guess this is a bit hand waving but I use the quotient law to get 5/0, so the limit dne. Where is the problem? How can I get into trouble doing this?
 
And if I get 0/0 then I try to either remove the 0/0 or I use L'Hopital's rule.
 
Jomo said:
If after direct substitution you get 7/0, you are done. I think I am right but I took calculus 1 back in 1986 (just over 30 years ago) and haven't taught it (or anything) for 11 years.
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I guess it depends on whether "limit exists" includes the case of infinity.
 
Jomo said:
I am not clear what you are saying. Suppose we have lim⁡x−>5xx−5\lim_{x->5}\dfrac{x}{x-5}. I guess this is a bit hand waving but I use the quotient law to get 5/0, so the limit dne. Where is the problem? How can I get into trouble doing this?
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Jomo said:
And if I get 0/0 then I try to either remove the 0/0 or I use L'Hopital's rule.
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Ultimately, the reason you can't use the rule as stated (even if the numerator were 5) is because they don't include that case in their statement of the theorem! If a theorem says "if d is not zero", and d is zero, you can't apply that theorem.

Also, the theorem as stated only lets you conclude that the limit does exist; it doesn't give you a way to conclude that it does not.

They could have written that theorem to include such a case, but it would have added complications: they would have to exclude the 0/0 case; and when the limit of the denominator is zero but the numerator is not, the way in which the denominator approached zero would make a difference in whether the limit is infinite or does not exist. Consider, for example, the limit of 1/x vs. 1/x^2. It's just a lot cleaner to make that a separate theorem or set of theorems.

There's a reason I asked for the book's statement of the theorem!

(By the way, I haven't actually taught calculus since before you learned it. That's one reason I'm cautious in helping students with it, even though I tutor in it frequently.)
 
Dr.Peterson said:
Ultimately, the reason you can't use the rule as stated (even if the numerator were 5) is because they don't include that case in their statement of the theorem! If a theorem says "if d is not zero", and d is zero, you can't apply that theorem.

Also, the theorem as stated only lets you conclude that the limit does exist; it doesn't give you a way to conclude that it does not.

They could have written that theorem to include such a case, but it would have added complications: they would have to exclude the 0/0 case; and when the limit of the denominator is zero but the numerator is not, the way in which the denominator approached zero would make a difference in whether the limit is infinite or does not exist. Consider, for example, the limit of 1/x vs. 1/x^2. It's just a lot cleaner to make that a separate theorem or set of theorems.

There's a reason I asked for the book's statement of the theorem!

(By the way, I haven't actually taught calculus since before you learned it. That's one reason I'm cautious in helping students with it, even though I tutor in it frequently.)
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So this question is not so much about the properties of mathematics but about applying the given theorem? They intentionally left out components of the theorem to make it all encompassing?
 
Integrate said:
So this question is not so much about the properties of mathematics but about applying the given theorem? They intentionally left out components of the theorem to make it all encompassing?
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Huh?

You can't use a property of mathematics until it has been proved. That is called a theorem. So a theorem is a property of mathematics!

But the problem you gave us explicitly says that it is asking about whether you can apply this theorem:
1611111504695.png

A particular theorem says what is true under certain circumstances. You can apply the theorem under those circumstances. You are being asked whether this limit fits those circumstances.

Nothing is "left out". Mathematics builds up from nothing (well, from axioms and definitions). Other circumstances are covered by other theorems (or by nothing at all, if those circumstances are useless). In this case, other theorems have to be applied, because "0/0" is too special a case to be included in a general theorem. An "all-encompassing" theorem would be too big and complicated to teach all at once.
 
Dr.Peterson said:
Huh?

You can't use a property of mathematics until it has been proved. That is called a theorem. So a theorem is a property of mathematics!

But the problem you gave us explicitly says that it is asking about whether you can apply this theorem:
View attachment 24516

A particular theorem says what is true under certain circumstances. You can apply the theorem under those circumstances. You are being asked whether this limit fits those circumstances.

Nothing is "left out". Mathematics builds up from nothing (well, from axioms and definitions). Other circumstances are covered by other theorems (or by nothing at all, if those circumstances are useless). In this case, other theorems have to be applied, because "0/0" is too special a case to be included in a general theorem. An "all-encompassing" theorem would be too big and complicated to teach all at once.
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I think I understand now. Thank you very much :)
 
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