Question

How do you find the limit of `(x/(x-3))` as x approaches `oo`?

Answer 1

`lim_(x->∞) (x)/(x-3) = 1`

Explanation:

This limit is also known as an infinite limit. In case of a polynomial such as this one, we can divide each term in the expression by the highest power of `x`.

`lim_(x->∞) (x)/(x-3) = lim_(x->∞) (x/x)/(x/x-3/x) = lim_(x->∞) (1)/(1-3/x)`

Since we know that `lim_(x->∞) 1/x = 0`, we can now solve this last limit.

`lim_(x->∞) (1)/(1-cancel(3/x)) = (1)/(1-0) = 1`

By graphing the function itself, we can also determine that the value approaches `1` as shown below.

graph{(x)/(x-3) [-10, 10, -5, 5]}