Easy math question???

Question

When you have 0 divided by 0, is the answer 0 or undefined???

I got this problem from a calculus book:

Find the limit of x/tan(x) as x approaches 0

Answer 1

Using calculus one could show that the answer approaches any possible answer (0, infinity, or anything inbetween), it depends on "how" the 0/0 is obtained.

For example x/x^2 as x approaches 0 gives a 0/0 that approaches infinity, but x^2/x as x approaches 0 gives a 0/0 that approaches 0.

Because of this the answer is usually stated as "undefined"


Using the additional details you added.

x/tan(x) as x goes to 0. The answer is 1.

Use L'Hopital's rule. take the derivative of the top and divide it by the derivative of the bottom and plug in 0.

The derivate of the top is 1.

The derivative of the bottom is d(tan(x)/dx = 1-sin(x)/cos(x)^2. Plugging in x = 0 gives 1.

Therefore you get 1/1 = 1.

Answer 2

You see in a limit you can't just plug in the values and try to wrap your head around an undefined value. Why is the value undefined? Because it changes with respect to the fractional functions.
Considering your question,
lim x / tan (x) = lim x.cos(x) / sin (x) = lim cos (x) = 1
x->0 x->0 x->0

This is because it can be shown, by calculus or geometry that
sin(x) / x = 1
=> x / sin (x) =1, when and only when x->0

and of course cos 0 =1

This is also evident from the MacLaurin/Taylor expansion of the function f(x) = x / tan (x)
the first term is 1, and as x->0 all the others become zero and unity is the answer.

Else differentiate the top and bottom seperately and consider that ratio which is the same as the original so long as the top and bottom are zero(or perhaps infinity). The proof is rather elementary though the applications of this powerful tool are often understated.
then the limit becomes,
lim 1/ sec^2 x = cos^2 (0)=1
x->0

Hope this helps!

Answer 3

Its undefined because of several reasons,

1)No number can be divided by 0 according to standard mathematic theory, but on the other hand general logic tells you that 0 will go into any number, other than its self, an infinite number of times.

2)General logic also states that 0 will go into it's self 1 time, but standard mathematics state that any number divided into 0 = 0.


So we are faced with a paradox being that 0/0 = 0 or 1 times.

0/0 = 1 false, nothing divided by nothing can’t equal something
0/0 = 0 false, a value divided by it’s self equals 1.

So being that there is are 2 impossible answers to this defined equation it would have to be undefined because you have two answers which are false.

Answer 4

0 divided by 0 is indeterminate. For example, we have C=0/0. That means 0*C=0. C can be any number.

It is NOT undefined. Undefined means that there is no answer. There are infinite possibilities for 0/0.

Btw, limit (as x approaches 0) of x/tanx can be solved using l'hopital's rule.
Because the limit is 0/0 (indeterminate!), you can differentiate numerator and denominator, to get
limit (as x approaches 0) of 1/(sec^2)(x).
Plug in x=0 and the answer is 1.

Answer 5

You need to understand the question and not get hung up on seeing zeros being divided by zeros.
Zero divided by zero is unsolvable, that is why we use limits to solve the problem of the tan of (x) as it approaches zero...
It never gets there...the tiniest number that is greater than zero is not zero...that is the whole point of limits, well half of the point anyway...the other half is that sometimes limits approach infinity because infinity also creates impossibilities.
In the case of imaginary numbers where i = the square root of -1 there is another use of an abstract concept to solve a previously unsolvable problem. (you can't take the root of a neg number)
Abstract concepts can be hard to accept...in the case of limits...they are an abstract concept that is applied to solve a previously unsolvable problem...go Newton!

The faster you can accept abstract concepts and move on the faster you can get your work done and get through the material.

Answer 6

Division by zero is undefined. Any number divided by zero produces an undefined quotient which, no matter what it is, cannot be multiplied by the divisor to produce the dividend.

Zero divided by 4 (or any rational number other than zero), however, is ok to do. The quotient is zero. Multiply the quotient, by the divisor, and you get the dividend.

Terms:
dividend: the number to be divided
divisor: the number you are dividing by
quotient: the anwer to a division problem

Answer 7

Undefined.

Answer 8

x/tan(x) as x---->0 is undefined. You can use L'hospital's rule. take the deriviative of the numerator & dividy by the derivative of the denominator. If this is defined, it is the answer. If not, continue this untill you get something defined.

Answer 9

Undefined. What number divided into zero makes zero? None.

Answer 10

0. cuz its basically: nothing divided by nothing. you remain with nothing. thus 0. sames goes for 0x0=0 (not undefined)