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What is 1 divided by 0?

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What is 1 divided by 0? Is it infinity?


Contrary to popular opinion, 1 divided by 0 is not infinity! Wikipedia states that “the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so division by zero is undefined“.

How to show that division by zero is undefined

$latex displaystyle lim_{xto 0^+} frac{1}{x}=+infty$

The limit of 1/x as x approaches zero from the right is positive infinity.

However, $latex displaystyle lim_{xto 0^-} frac{1}{x}=-infty$

The limit of 1/x as x approaches zero from the left is negative infinity.

Since the left limit and right limit are different, the limit of 1/x as x approaches infinity does not exist!

Note: There are mathematical structures in which a/0 is defined for some a (see Riemann sphere, real projective line, and section 4 for examples); however, such structures cannot satisfy every ordinary…

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Do They Get It? The Instantaneous Rate of Change Exactly

Continuous Everywhere but Differentiable Nowhere

Today in calculus I wanted to check if students really understood what they were doing when they were finding the instantaneous rate of change. (We haven’t learned the word derivative yet, but this is the formal definition of the derivative.)

So I handed out this worked out problem.

And I had them next to each of the letters write a note answering the following individually (not as a group):

A: write what the expression represents graphically and conceptually

B: write what the notation $latex \lim_{h\rightarrow0}$ actually means. Why does it need to be there to calculate the instantaneous rate of change. (Be sure to address with h means.)

C: write what mathematical simplification is happening, and why were are allowed to do that

D: write what the reasoning is behind why were are allowed to make this mathematical move

E: explain what this number (-1) means, both conceptually and graphically

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Practical Uses for Calculus

The question finally escaped from the lips of one of my precalculus students.  The conversation went something like this:

Student: What is all this stuff useful for anyway?

Me: Precalculus is useful for preparing you for calculus.

Student: No, I mean what is calculus useful for?  Do we ever really use calculus for real life situations?

Me: Of course!  One example is the headlight used in cars.  Calculus was used to help maximize the light emitted from the headlight.

Student: As long as it’s useful for something.

He seemed satisfied.  I smiled.

As he walked away, I thought about the numerous times my students ask about the purpose of math, but this was a first.  This student wanted to know that calculus was good for something and useful in the “real world.”  So I did a little “research” and posted several sites in case someone reading this blog shares the same curiosity.

Some uses for calculus (very brief):

1. Growth rate of tumors (medicine)

2. Minimum payments due on credit cards (finance)

3. Curves for bridges, tunnels, and more (engineering)

4. Area under a curve (engineering)

5. Marginal cost (economics)

Websites to visit for more details (just a few):

NOTE: Photo borrowed from:

What are some other practical uses for calculus?