How do you use the formal definition of a limit to prove #lim (x^2-x+1)=1# as x approaches 1?
2 Answers
Any polynomial f(x) with real (single valued) coefficients is single valued. The quadratic has the value 1 at x= 1. Only for discontinuous fuctions, we have limiting values for different approachesi
Explanation:
The question of passing on to the limit, if, any, does not arise for continuous single-valued functions. The quadratic is a continuous and single-valued function..
In general, the formal way for finding the limit is to substitute x = 1 + h and take the limit as
See below.
Explanation:
Preliminary work
We need to show that
for any positive
For every
Note that
We start by putting an initial bound on
With this initial requirement, we see that
Now if we ALSO make sure that
Here then is the proof :
Given
Now if
a)
#absx < 3# , andb)
#abs(x-1) < epsilon/3#
So
This completes the proof.