Question

How do you find the limit of `(x^2-1)/(x-1)` as `x->1`?

Answer 1

`lim_(x rarr 1)(x^2-1)/(x-1) = 2`

Explanation:

Let `f(x) = (x^2-1)/(x-1)` then `f(x)` is defined everywhere except at `x=1`, however when we evaluate the limit we are not interested in the value of `f(1)`, just the behaviour of `f(c)` for `c` close to `1`.

`lim_(x rarr 1)(x^2-1)/(x-1) = lim_(x rarr 1)((x+1)(x-1))/((x-1))`
`:. lim_(x rarr 1)(x^2-1)/(x-1) = lim_(x rarr 1) (x+1), " as " x!= 1`
`:. lim_(x rarr 1)(x^2-1)/(x-1) = 1+1`
`:. lim_(x rarr 1)(x^2-1)/(x-1) = 2`

graph{(x^2-1)/(x-1) [-10, 10, -5, 5]}