Over the given interval

I

determine the sets of uniform convergence for:

f

n

(x)=

x

1+n

x

2

,I=R.

Hint: To find the suprimum of

g

n

(x)=|

f

n

(x)−f(x)|

on

R,

let

t=|x|

for

t≥0.

Another trick, let

u=

n

−

−

√

|x|

and use the inequality

u

1+

u

2

≤

1

2

for all

u∈R.

Question 2

Select one:

a.

f

⟶

uniform

0

on

R.

b.

f

⟶

uniform

0

on

[−1,1].

c.

f

⟶

uniform

0

on

[0,1].

d.

f

⟶

uniform

0

on 

We are asked to determine the sets of uniform convergence for the sequence of functions \( f_n(x) = \frac{x}{1 + nx^2} \) over the interval \( I = \mathbb{R} \). The goal is to identify the interval on which \( f_n(x) \) converges uniformly to 0.

### Step 1: Pointwise Convergence
First, let's determine the pointwise limit of \( f_n(x) \) as \( n \to \infty \).

\[ f_n(x) = \frac{x}{1 + nx^2} \]

As \( n \to \infty \), the term \( nx^2 \) dominates in the denominator if \( x \neq 0 \), so:

\[
\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{x}{1 + nx^2} = \lim_{n \to \infty} \frac{x}{nx^2} = \lim_{n \to \infty} \frac{1}{nx} = 0
\]

Thus, \( f_n(x) \) converges pointwise to 0 for all \( x \in \mathbb{R} \).

### Step 2: Analyzing Uniform Convergence
To determine whether this convergence is uniform, we analyze the difference \( |f_n(x) - 0| = |f_n(x)| \) and find the supremum of this expression over \( \mathbb{R} \).

We are given a hint to let \( t = |x| \) and \( u = \sqrt{n}|x| \). Rewriting the function in terms of \( u \), we have:

\[
g_n(x) = |f_n(x) - 0| = \left|\frac{x

Explore the uniform convergence of the sequence of functions f_n(x) = x / (1 + nx^2) over the interval I = R. Understand how pointwise convergence leads to the determination of sets where f_n(x) converges uniformly to 0. Suitable for undergraduate-level mathematics.

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