Limit of (x^2 + x) / (x^5 + 2x^4 + x^3) as x -> 0

[hank]

The problem is as follows:

Find the limit as x->0 of (x^2 + x) / (x^5 + 2x^4 + x^3)

Here's my solution:

= lim x->0 [x(x+1)] / (x^3[x^2 + 2x + 1])
= lim x->0 [x(x+1)] / (x^3(x+1)(x+1))
= lim x->0 1 / (x^2(x+1))

This is where I get stuck.

If I direct sub 0 into x, the equation should be undefined, shouldn't it? However, the answer is infinity.

That leaves me to believe that the answer ends up being lim x->0 of 1/x^2 which, according to the graph, would make the limit infinity.

However, subbing in 0 for x makes it undefined. What am I missing here?

Thanks..

--Hank

 

[skeeter]

What am I missing here?

the entire idea behind a limit ...

yes, at x = 0, the function is undefined ... but when determining a limit, we don't care what the function is at x = 0, we care how the function behaves as x gets close to 0.

as x->0 the function in question "blows up" or exhibits unbounded growth toward infinity.