Question

Explain why:

lim x-> -infinity 1 / (4^x + 1)

= 1 ?

Answer 1

I can simplify it down to:

\[\frac{ 1 }{ \infty + 1 }\]

Doesn't this equate to 0?

Answer 2

Okay, for this problem we need to break it down into two parts.

First... think about the Lim of 4^x

We can easily see that as x gets larger, 4^x gets even larger. What's hard to visualize is when x becomes negative. For this kind of problem, I'd either graph it, or plug a few problems into my calculator.
4^0 = 1
4^(-1) = 1/4
4^(-2) = 1/8
4^(-3) = 1/16
So as x goes to negative infinity... 4^x gets really small.. in in short. Zero.

So really what you have is \[\frac{ 1 }{ 0 + 1 }\]

Does that help?

Answer 3

The trick is when you see a negative power, think of it as it raised to the inverse power.

\[4^{-x} = \frac{ 1 }{ 4^x }\]