Explain what is a limit with an example
The limit y = f(x) at x = a is the value taken on by y when x approaches a but is not actually equal to a.
For a function f(x) that is not be defined at x = a, y = f(a) does not return a valid value but `y = lim_(x->a) f(x)` may return a valid value as it is what f(x) is equal to when x is infinitesimally close to a.
It should be kept in mind that it is possible to make x equal to to a value x' that is closer to a than x is without x' actually being equal to a.
For example: consider `lim_(x->0) (sin x)/x` . `sin 0/0 = 0/0` which is a value that cannot be determined. At x = 0, the expression does not have a valid value. The limit `lim_(x->0) (sin x)/x` gives the value taken on by `(sin x)/x` as x is infinitesimally close to 0 though not equal to it.
Limit of a function (y) with x tend to a is defined as the a function which y values varies when x approces a but it si not equal to a
now lim (x->5) y = x^2 - 5
here the value of y will varies as x approes to 5 but not equal to 5.
to solve this we subsititute the value of x as 5
that is y = 5^2 - 5 =25-5 =20
so the limit of the funtion is 20 that is it can be 19.99999999 but not exactly 20.