Question

B
asically what is the main idea behind 'limits'?

Answer 1

The process of taking the limit of a function is approximating the function value as x approaches some value (e.g. a).
\[\lim_{x \rightarrow a}F(x) = L\]

Answer 2

you can get very close to some value

Answer 3

a limit is defined basically as getting closer to the same number from the left as you do from the right

Answer 4

nayeaddo limits are not approximations.

Answer 5

the reason we have to get close to, and not care about the value at the limit, is becasue we tend to want to know the value for an indeterminate number such as 1/0

Answer 6

@ Alchemista you are right I meant to use the word approaching

Answer 7

A limit is equivalent to stating that the following holds
\[\lim_{x \to a} f(x) = L\]If for every \(\varepsilon > 0\) there exists a \(\delta > 0\) such that when \( 0 < |x - a| < \delta\) then \(|f(x) - L| < \varepsilon\)

Answer 8

Always keep in mind though that the limit of a function as you approach a value can be quite different from the solution *at* that value.

Limits are only concerned with what happens slightly *away* from the x-value.

If the limit equals the function value at the x-value, you have continuity :)

Answer 9

Thanks for explanation

Answer 10

The reason the "getting closer" language is not precise is because that's not always true.

For instance with suppose you have a constant function \(f(x)=c\), then for any \(a\)
\[\lim_{x \to a}f(x)=c\]For every \(\varepsilon > 0\) you can take any \(\delta>0\) and the statement will hold. But while the limit holds it doesn't "get closer" in the way you described.

Answer 11

There are certain situations where the values keeps approaching a specific value but never attains it.
consider,
1/x

put different values for x,
x=1,2,3,4,.......10000.....1000000.......then
1/x will be 1,0.5,0.333,0.25,......0.0001,.....0.000001........
It appears as if the numbers are moving towards zero but you'll never have the value zero by dividing 1 by any value.
here zero is the limit