With the limits as x approaches some number c, then Most of the time, if you have two functions f and g with The word "indeterminate" comes up in the context of limits of functions of real variables, which is something that you learn about usually in a first calculus class. There is no number called "indeterminate", and there is no commonly used number system where multiplying two elements together yields something called "indeterminate". The idea that 0*∞="indeterminate" is a common misconception.
When it is defined, it is usually defined to be zero. But most of the time ∞*0 is left undefined, making it as meaningless a phrase as it is in the real numbers. Limits of real valued functions are extended-real-valued, meaning usually we ∞ and -∞ to be in the set of allowable values of the expression lim f(x). There is another set called the extended real numbers, which includes elements ∞ and -∞. So in the real numbers, "infinity times zero" is meaningless. The set of real numbers do not include an element called "infinity", and so if we are dealing with real numbers, "infinity times zero" has about as much meaning as "elephant times zero". And that leads you to transforming it into one of the appropriate forms like 0/0 or infinity/infinity and then applying L'Hôpital's rule.įirst thing's first: "infinity times zero" is a meaningless phrase in most usual contexts. For example, if f(x) approaches infinity faster than g(x) approaches 0 then why would f(x)*g(x) approach 0? It will get larger in value, not smaller. We are worried about the values of x before it. We aren't worried about the product of f(0)*g(0). But why? f(x) approaches infinity as x approaches 0. We have lim of f(x) g(x) as x approaches 0 = infinity0 = 0. Let's put it another way and apply your reasoning. You are trying to figure out what that product approaches as x approaches some value and you've gotten to the point that substitution no longer works because the product is undefined.
In other words, there is no value of x that produces infinity*0 as the product of f(x)*g(x). The key thing to understand in regards to your question is that with the limit you are never actually multiplying anything by infinity or anything by 0 in a limit. You may be able to transform it into another form and apply L'Hôpital's rules and find the limit. It means that the limit cannot be determined from the form used because an actual limit cannot be produced. This doesn't mean the limit is "undefined" or that the limit doesn't exist. After substitution in a limit you can arrive at an indeterminate form like infinity * 0. Outside of the context of limits infinity * 0 would just be undefined for the reasons people have stated, multiplication is defined for numbers and infinity is not really a number. This is why 0 x infinity is undefined, we can't assign a number to it that is consistent with all our observations. So I can make (0 x infinity) be anything I want. In fact, look at the function g(x)=ax(1/x), "plugging in" x=0 gives (0)(infinity), but if you look at limits lim g(x) as x->0 is a. What happens when we have 0 x infinity? What happens when an unstoppable force meets an immovable object? You can find times when it "should" be 5, or 4, or any other number. That being said, 0x(anything)=0 and infinity(anything)=infinity. If anything, 1 is halfway between 0 and infinity. So the number smack in the middle of 0 and infinity should be a number that is not changed by the transformation given by f(x)=1/x. In fact, f(0)=infinity and you could say that f(infinity)=0. In fact, if you look at the function f(x)=1/x, this takes large numbers, makes them small and takes small number, makes them big. But we can recognize this duality between 0 and infinity. Any finite number is a bad measure for the size of a number. The point is, 10 is a bad measure for the size of a number. It's infinitely bigger than zero! I had a professor who would joke "Let's choose an exceptionally large number, 10 -100". You could say the same thing about infinity as you do zero: Any number times infinity is infinity: infinity x 5= infinity for sure. I had a professor who would joke "Let's choose an exceptionally small number, 100 Trillion". Any number that you can think of if infinity smaller than infinity. 10 1000001000000010000000 is a very large number, compared to 10, but compared to infinity it's stupid small.
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