Question

Prove the following limit statement:

lim (X^2-1)/(x-1) = 2
x-->1

Answer 1

By using the precise definition of a limit.

Answer 2

then let's let \( \varepsilon > 0 \) and be done with it ^.^

Answer 3

sorry @OneMathCat please carry on ^_^

Answer 4

First of all, note that (x^2-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1, providing x is not equal to 1.
Thus, as x approaches 1, x+1 --> 2. So, surely sounds like it's true!
Now, for the precise definition, you must show that for all epsilon > 0,
there exists delta > 0, such that if 0 < |x-1| < delta,
then | (x+1) - 2 | < epsilon.
Note that | (x+1) - 2 | = | x - 1| .
I'll let you take it from here! Good luck!
(So think --- how should you choose delta?)