Limit of Unknown Function: If lim x→3 of (x+10)/f(x) = 0, find lim x→3 f(x).

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invasionmoo

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Limit of Unknown Function: If lim x→3 of (x+10)/f(x) = 0, find lim x→3 f(x).

Question: If lim x→3 of (x+10)/f(x) = 0, find lim x→3 f(x).

My answer was the lim x→3 f(x) = ∞ because anything divided by ∞ would approach 0. However, my teacher told me I was wrong. Does anyone know what I did wrong?
 
invasionmoo said:
Question: If lim x→3 of (x+10)/f(x) = 0, find lim x→3 f(x).

My answer was the lim x→3 f(x) = ∞ because anything divided by ∞ would approach 0. However, my teacher told me I was wrong. Does anyone know what I did wrong?
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What reason did the teacher give for why your answer was wrong? What was the exact wording of the problem?

I do not love your answer because it says nothing about other possible values of f(x). I'd prefer an answer that says that the limit of a quotient is equal to the quotient of the limits unless the limit of the denominator is zero or the limits of both numerator and denominator are infinite. In this case the limit of the numerator is 13, and no finite limit of the denominator will result in a quotient of zero.Therefore, the only possible limit of the denominator is infinity.

If that was your teacher's reasoning, that makes sense to me. But I would not say that you were wrong. I would have said that your answer was incomplete.
 
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invasionmoo said:
… My answer was the lim x→3 f(x) = ∞ … my teacher told me I was wrong.
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The exercise tells us that (x+10)/f(x) approaches 0 as x approaches 3, but it doesn't say whether (x+10)/f(x) approaches 0 from above or from below. Consider this (i.e., two possibilities exist) and how it changes your answer.

Personally, I would report that f(x) has no limit, as x approaches 3. In other words, I would say the limit does not exist. Every limit is a number. (Infinity is not a number.)

When mathematicians write or say limit f(x) = ∞, that's an abbreviation. It means: "The limit does not exist because f(x) increases without bound."

Likewise, when they write or say limit f(x) = -∞, that's also an abbreviation. It means: "The limit does not exist because f(x) decreases without bound."
 
invasionmoo said:
anything divided by ∞ would approach 0. However, my teacher told me I was wrong. Does anyone know what I did wrong?
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If your teacher said that the statement above is wrong, then I have to agree with your teacher as ∞/∞ does not necessarily = 0. In fact if you give me any real number k or +/- ∞ I can make up a limit that give ∞/∞ but equals k.

Consider lim x→3 f(x) = (k/(x-3))/(1/(x-3)) = ∞/∞ = k
 
mmm4444bot said:
I'm thinking the teacher told invasionmoo the answer is wrong, not neccesarily the reasoning.
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I probably agree with you (I did not give it much thought). I just wanted to point out that anything divided by infinity is not 0.
 
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