May I ask what level (of school/university) is this topic for?
(Last time I did this what perhaps 40+ years ago. but I remember something like this:)
If a function f is defined by f(x) = (x*x - 1)/(x - 1); let us find a limit of f(x) at x = 1.
If we are are told x = 1. We substitute and get f(1) = (1*1 - 1)/(1-1) = (0)/(0) and we don't have a definition for division by 0 (zero). So finding limits of some functions can be tricky.
Now if we tried increasing value of x say from 0.5 to 0.9 and decreasing value of x say from 1.5 to 1.1, (work them out here), we should see that f(x) is approaching 2.0 as x approaching 1.0. The limit of f(x) = (x*x - 1)/(x - 1) is 2. Not indeterminate (0/0)!
Alternatively, if we factor out (x*x - 1) to (x + 1)(x - 1) then f(x) = (x+1)(x-1)/(x-1)=(x+1) and at x =1 we have f(x)=2.
What is the idea of "continuity of a function"? We say a function f is continuous (within a range) when f has a value at any value x and at x+s for any small s. Logically, if we can find value of f(x) at x and at any value of x +/- small increment s for any s (as small as we think of "the next point") then the function f is continuous (or f(x) has values for all values of x).
May I ask what level (of school/university) is this topic for?
(Last time I did this what perhaps 40+ years ago. but I remember something like this:)
If a function f is defined by f(x) = (x*x - 1)/(x - 1); let us find a limit of f(x) at x = 1.
If we are are told x = 1. We substitute and get f(1) = (1*1 - 1)/(1-1) = (0)/(0) and we don't have a definition for division by 0 (zero). So finding limits of some functions can be tricky.
Now if we tried increasing value of x say from 0.5 to 0.9 and decreasing value of x say from 1.5 to 1.1, (work them out here), we should see that f(x) is approaching 2.0 as x approaching 1.0. The limit of f(x) = (x*x - 1)/(x - 1) is 2. Not indeterminate (0/0)!
Alternatively, if we factor out (x*x - 1) to (x + 1)(x - 1) then f(x) = (x+1)(x-1)/(x-1)=(x+1) and at x =1 we have f(x)=2.
What is the idea of "continuity of a function"? We say a function f is continuous (within a range) when f has a value at any value x and at x+s for any small s. Logically, if we can find value of f(x) at x and at any value of x +/- small increment s for any s (as small as we think of "the next point") then the function f is continuous (or f(x) has values for all values of x).
Now maths can be fun ;-)