Please pay attention This is not a calculus course rather it

Please pay attention. This is not a calculus course rather it is a Math Proof course. So proving using calculus like when C approach C+ and C- as h approaches 0 will not work

Let a, b, and c be real numbers. Prove that limxc (ax + b) = ac + b.

Solution

Let f(x) = ax+b

Let us substitute x = c+h where h is very small

Then f(x)-f(c) = f(c+h)-f(c) = a(c+h)+b-(ac+b) = ah

Similarly for x = c-h,

f(x)-f(c) = f(c-h)-f(c) = a(c-h)+b-(ac+b) = -ah

Or in other words we see that

whenever |x-c|<h, |f(x)-f(c)|<ah

In other words limit of f(x) as x tends to c is f(c) = ac+b