- #1
Mr Davis 97
- 1,462
- 44
I am confused about the algebraic process of finding a limit. Let us take ##\frac{x^2 -1}{x - 1}##. In trying to find ##\lim_{x\rightarrow 1}\frac{x^2 -1}{x - 1}##we do the following:
##\displaystyle\lim_{x\rightarrow 1}\frac{(x+1)(x-1)}{x - 1}##
##\displaystyle\lim_{x\rightarrow 1}x+1##
##2##
But what justification do we have for cancelling the (x - 1) terms? When we cancel these terms, we are effectively dealing with a whole new function since the domain changes. Why is changing functions allowed in doing limits? How can we be absolutely certain that ##\lim_{x\rightarrow 1}x+1## leads to the correct answer to the original problem if ##x + 2## is different function, with a different domain, than ##\frac{x^2 -1}{x - 1}##?
##\displaystyle\lim_{x\rightarrow 1}\frac{(x+1)(x-1)}{x - 1}##
##\displaystyle\lim_{x\rightarrow 1}x+1##
##2##
But what justification do we have for cancelling the (x - 1) terms? When we cancel these terms, we are effectively dealing with a whole new function since the domain changes. Why is changing functions allowed in doing limits? How can we be absolutely certain that ##\lim_{x\rightarrow 1}x+1## leads to the correct answer to the original problem if ##x + 2## is different function, with a different domain, than ##\frac{x^2 -1}{x - 1}##?