Do you understand that the following is equal to 1?
I have a little question. It is fact that:
But when I have a limit:
it is considered an "indeterminate expression", because it is equal to:
but I think that: and it would be also an indeterminate expression.
So can anybody explain me why "is the first expession = 1"?
Some instructor has given you a great deal of misinformation.
Did you know that when evaluating limits such as it is the case that never, never, never equals !
So in the limit it is true that never, never, never equals 0.
So if then so that is the limit.
In the other limit for any (close to zero but not equal to zero) it is true that so it follows that .
Look up L'Hopital's Rule:
If you have an intermediate situation like sin(x)/x, lim x appr. 0, take the derivative of the numerator and the denominator independently:
d/dx(sin(x)) = cos(x)
d/dx(x) = 1
So your solution is:
cos(x)/1, lim x appr. 0
Since cos(0)=1, you have:
What your teacher meant with "indeterminate expression" is the following.
If you only know about a fraction, that the numerator and the denominator both approach 0, then the limit of the quotient could be anything (+/- infinity or 0 or 32 or 1347.312 or any other real number)
Knowing that it's in the form 0/0 is no information about it's limit.
It is indeterminate in a sense that the 0/0 form does not determine the limit of the fraction like for example 2/3 does (if the numerator approaches 2 and the denominator approaches 3, the whole fraction approaches 2/3 for sure).