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negation
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Homework Statement
Suppose the lim x-> a f(x) = + infinity and lim x->a g(x) = 0
then why would lim x→a (f(x)×g(x)) be insufficient to tell us anything about the product of 2 limit?
negation said:Homework Statement
Suppose the lim x-> a f(x) = + infinity and lim x->a g(x) = 0
then why would lim x→a (f(x)×g(x)) be insufficient to tell us anything about the product of 2 limit?
negation said:Homework Statement
Suppose the lim x-> a f(x) = + infinity and lim x->a g(x) = 0
then why would lim x→a (f(x)×g(x)) be insufficient to tell us anything about the product of 2 limit?
LCKurtz said:Look at these two examples taking ##a=0## so ##x\rightarrow 0^+##:
##f(x) = \frac 1 x \rightarrow \infty,~g(x) = x^2\rightarrow 0##. Here ##f(x)g(x) = x\rightarrow 0##
Now take ##f(x) = \frac 1 {x^2}\rightarrow \infty,~g(x) = x\rightarrow 0##. Here ##f(x)g(x)= \frac 1 x\rightarrow \infty##.
So you having ##f\to\infty,~g\to 0## isn't sufficient to tell us anything about ##fg##.
Equal what?negation said:This is tough to grasp or maybe I'm missing some intermediate steps.
By the basic limit law of multiplication:
lim x->a f(x) and lim x->a g(x)
The property you are citing, about the multiplication of limits, requires that both limits exist. That means that each limit has to be a finite number. So your first limit does not exist.negation said:then lim x->a f(x) . lim x->a g(x) = lim x->a f(x) .g(x)
No.negation said:so f(x) -> infinity and g(x) -> f(x).g(x) = 0
Why complicate things by mixing up a and zero?HallsofIvy said:And to complete what LCKurtz said, for a any non-zero number,
If [itex]f(x)= \frac{a}{x}[/itex] and [itex]g(x)= x[/itex], then [itex]\lim_{x\to 0} f(x)g(x)= a[/itex]
So that, in fact, there are examples giving every possible result!
I don't know what you mean by this. Mixing up "a" and what "zero"?oay said:Why complicate things by mixing up a and zero?
HallsOfIvy is making the point that an [∞ * 0] indeterminate form can turn out to be any number.oay said:Why complicate things by mixing up a and zero?
I believe there was only one zero in your post - that zero was the one you apparently don't see as the problem.HallsofIvy said:I don't know what you mean by this. Mixing up "a" and what "zero"?
Thanks, I was aware of that.Mark44 said:HallsOfIvy is making the point that an [∞ * 0] indeterminate form can turn out to be any number.
Limits in calculus refer to the value that a function approaches as the input approaches a certain value. It is used to study the behavior of a function near a given point.
Limits are important in calculus because they help us understand the behavior of a function and make predictions about its values. They also play a crucial role in the definition of derivatives and integrals.
To evaluate a limit, you can use direct substitution, factoring, or algebraic manipulation. You can also use graphical methods or L'Hopital's rule in certain cases. Practice and identifying the type of limit will help you develop your limit-evaluating skills.
The common types of limits include limits at a point, limits at infinity, one-sided limits, and limits involving trigonometric, logarithmic, and exponential functions.
Yes, there are some special techniques such as using trigonometric identities, rationalizing the numerator or denominator, and using substitution or squeeze theorem. It is important to understand the properties of limits and practice solving various types of limits to develop your problem-solving skills.