# Thread: Sequence of Functions Convergence

1. ## Sequence of Functions Convergence

I'm not too sure how to do this, so any help would be great:

fn(x) = n/(1+nx^2) on (0, infinity)

a) find the pointwise limit
b)prove (or disprove) if the sequence converges uniformly.

I don't really know how to find the pointwise limit because I can't factor the denominator, and I can't separate the numerator into two fractions (like the examples in the book).

Also, I have another one that is a bit different:

fn(x) = sin(nx)/1+nx on [1,2].
How do I figure out the pointwise limit on that domain?

2. $\frac{n}{1+nx^2} = \frac{n}{n} \cdot \frac{1}{\frac{1}{n}+x^2} = \frac{1}{\frac{1}{n}+x^2}$.

Hence $\frac{n}{1+nx^2} \longrightarrow \frac{1}{x^2}$ pointwise.

3. $-\frac{1}{1+nx} \leq \frac{\sin(nx)}{1+nx} \leq \frac{1}{1+nx}$.

Now apply the squeeze theorem.