Can We Use the Product Rule for Limits in This Scenario?

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In summary: Basically what you're saying is that if a function is bounded on the left and right, then the function's limit does not exist.
  • #1
MHD93
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Note: all the following limits are when x approaches some certain number

The question is: if lim(f(x)g(x)) doesn't exist, and lim(f(x)) = 1, and lim(g(x)) doesn't exist
then are we allowed to say that lim(f(x)g(x)) = lim(f(x)) * lim (g(x)) = 1 * lim(g(x)) = doesn't exist

or isn't that allowed in general

Thanks alot
 
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  • #2
if lim (f(x)) is continuous and lim (g(X)) is not then it follows that:
lim(f(x)*g(x)) is not continous...for the simple reason that f(x) is bounded and g(x) is not bounded...for e.g if f(x)= 1/n then its and n→∞ then the lim→0 and conversely for g(x)=n when n→∞ lim→∞
 
  • #3
What you're saying is allowable, but I want to tell you that it's useless and makes absolutely no sense to repeat yourself 3 times. I'm not a pro on limits but you should understand what you're doing...

The simple reason is because all your doing is stating what is equal three times

lim(f(x)g(x) = lim(f(x)) *limg(x)) All you did was show us another way to write how to multiply a function

then you say that 1 * lim(g(x)) =doesn't exist

Of course not

if f(x) = 1 all you did was plug in a 1 instead of F(x)

You mine as well just said

lim(f(x)g(x)) = lim(f(x)) * lim (g(x)) = 1 * lim(g(x)) = limf(x) * limg(x) = lim(f(x)g(x))
^ If you would've just worked your way back around you would've made it.

Essentially all you did was substitution, but yeah not really a limit question more of a algebra question
 
  • #4
SpeedOfDark said:
What you're saying is allowable, but I want to tell you that it's useless and makes absolutely no sense to repeat yourself 3 times. I'm not a pro on limits but you should understand what you're doing...

The simple reason is because all your doing is stating what is equal three times

lim(f(x)g(x) = lim(f(x)) *limg(x)) All you did was show us another way to write how to multiply a function

then you say that 1 * lim(g(x)) =doesn't exist

Of course not

if f(x) = 1 all you did was plug in a 1 instead of F(x)

You mine as well just said

lim(f(x)g(x)) = lim(f(x)) * lim (g(x)) = 1 * lim(g(x)) = limf(x) * limg(x) = lim(f(x)g(x))
^ If you would've just worked your way back around you would've made it.

Essentially all you did was substitution, but yeah not really a limit question more of a algebra question
spped,
was i right in my response...thanks and goodnight

chwala
 
  • #5
Thanks guys

but I want to tell you that it's useless and makes absolutely no sense to repeat yourself 3 times

If you just knew why I asked this questions, you wouldn't have said that
I have a limit in my exam which is lim( ((x^2 - 2x +1) ^ 0.5) / (x(x-1))) as x approaches one
in the end the limit doesn't exist, what I did is separate this to two limits, one of which is lim 1/x
and continued my solution properly
without this separation I would have taken a full mark in my exam, but my teacher said it's not allowable because one of these limits doesn't exist,
so he took from me a mark and had given me a big bang :D

I'm insisting to convince him so help me ;)

thanks
 
  • #6
The problem is, you're assuming what you want to prove.

Splitting up only works if you know that the limit already doesn't exist, in which case it defeats the purpose of the question. For example,

[tex]
1 = \lim_{x \rightarrow 0} 1 = \lim_{x \rightarrow 0} \frac{x}{x} = \lim_{x \rightarrow 0} x \cdot \frac{1}{x}
[/tex]

If you split it it, you get one of the factors that doesn't exist, but the limit itself exists. You have to be more careful when making such algebra tricks, which is why your prof took off points. Too careless. : )
 
  • #7
chwala said:
if lim (f(x)) is continuous and lim (g(X)) is not then it follows that:
lim(f(x)*g(x)) is not continous...for the simple reason that f(x) is bounded and g(x) is not bounded...for e.g if f(x)= 1/n then its and n→∞ then the lim→0 and conversely for g(x)=n when n→∞ lim→∞

Hey chwala.. probably u have to be more careful... u said f(x) has limit because it is bounded... now consider [tex](-1)^{n}[/tex] .. But I am sure what u've state about g(x) exisiting no limits before it is not bounded ..
 
  • #8
icystrike said:
Hey chwala.. probably u have to be more careful... u said f(x) has limit because it is bounded... now consider [tex](-1)^{n}[/tex] .. But I am sure what u've state about g(x) exisiting no limits before it is not bounded ..

thanks you are right...(-1)^n is bounded by -1 and 1 but has no limit i.e if n→∞ the limit does not exist...i withdraw the statement on the boundedness of f(x).
 
  • #9
l'Hôpital said:
The problem is, you're assuming what you want to prove.

Splitting up only works if you know that the limit already doesn't exist, in which case it defeats the purpose of the question. For example,

[tex]
1 = \lim_{x \rightarrow 0} 1 = \lim_{x \rightarrow 0} \frac{x}{x} = \lim_{x \rightarrow 0} x \cdot \frac{1}{x}
[/tex]

If you split it it, you get one of the factors that doesn't exist, but the limit itself exists. You have to be more careful when making such algebra tricks, which is why your prof took off points. Too careless. : )

mustn't there be a difference between a non-existing limit and an undefined limit?
 
  • #10
please help me?

if limit f(x)=L and limit g(x)app infinity then
xapp a xapp a


prove by c,delta methodthat (f(x)+g(x)) approaches infinity when x approaches a


thanks
 

What is a limit in mathematics?

A limit in mathematics is a fundamental concept that defines the behavior of a function as its input approaches a certain value. It is used to describe the value that a function approaches, but may not necessarily reach, as its input approaches a specific value.

What is the purpose of finding limits?

The purpose of finding limits is to understand the behavior of a function and its graph as its input approaches a certain value. This helps in determining the continuity, differentiability, and other important properties of a function.

How do you find limits?

Limits can be found by using various techniques such as algebraic manipulation, substitution, and L'Hopital's rule. It is also important to understand the properties and rules of limits in order to solve more complex problems.

What is the difference between one-sided and two-sided limits?

A one-sided limit only considers the behavior of a function as its input approaches a specific value from one side, either from the left or the right. A two-sided limit considers the behavior of a function from both sides as its input approaches a specific value.

What is the significance of limits in real-world applications?

Limits are used in various real-world applications such as in physics, engineering, economics, and other sciences. They help in modeling and predicting the behavior of systems and processes, and are essential in understanding the concept of continuity and change.

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