Question

How do you find the limit of `(x+3)^2006` as `x->-4`?

Answer 1

By directly substituting in the value.

Explanation:

We can substitute `-4` for `x` directly. Unlike other limit problems, the function is defined at `x=-4` so we can plug it right into the function.

`lim_(xrarr-4)(x+3)^2006=(-4+3)^2006=(-1)^2006=1`

Note that `(-1)^m=1` when `m` is even and `(-1)^n=-1` when `n` is odd. You can prove this to yourself by multiplying out a couple examples like `(-1)^3`, `(-1)^4`, and `(-1)^5`. In an even-powered example, each `-1` will be paired with another `-1`, leaving a positive result. With an odd power, there will always be one leftover `-1`.

You can also think about `(-1)^2006` as `((-1)^2)^1003=1^1003=1`.