Question

Letf_n(x) = \frac{nx}{1+nx^2}

a. Find the pointwise limit of $(f_n)$ for all $x \in (0, +\infty)$.

b. Is the convergence uniform on $(0, +\infty)$?

c. Is the convergence uniform on $(1, +\infty)$?

a-c Please

Answer #1

For each $n \in \mathbb{N}$ and $x \in [0, +\infty)$ let g_n(x)
= \frac{x}{1 + x^n}
a. Find the pointwise limit on $[0, +\infty)$
b. Explain how we know that the convergence \textit{cannot} be
uniform on $[0, +\infty)$
a & b Please

Find the pointwise limit of the following sequences of
functions on the segment [0,2]. Is this convergence uniform on
[0,2] a) fn(x)=(x^n)/((x^n)+1

\sum _{n=1}^{\infty
}\left(\frac{3}{2^n}+\frac{8}{n\left(n+1\right)}\right)

Calculate each limit below, if it exists. If a limit does not
exist, explain why. Show all work.
\lim _{x\to 3}\left(\frac{x-3}{\sqrt{2x+3}-\sqrt{3x}}\right)
\lim _{x\to -\infty }\left(\frac{\sqrt{x^2+3x}}{3x+1}\right)

$\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+....+\frac{1}{n!}<3-\frac{2}{(n+1)!}$\
$\forall\in\mathbb{N}$ / {1} proof by induction

Part b: Find the limit as x goes to zero for
((1/x)-cosx/sinx))
part c: find all values for c and d for which the limit as x
goes to 0 for [(c+cos(dx))/x^2]=-2
please prove

Find the limits, or state that the limit does not exist (you
must justify answer):
a) \lim_(x->\infty )(\sqrt(x^(2)+2x)-\sqrt(x^(2)-x))
b) \lim_(x->\infty )(lnx-ln(sin x))
c) \lim_(x->\infty )x^((1)/((lnx)))

Consider the following limit.
lim (x^2 + 4)
(x--> 5)
1. Find the limit L.
2. Find the largest δ such that |f(x) −
L| < 0.01 whenever 0 < |x − 5| < δ.
(Assume 4 < x < 6 and δ > 0. Round your answer to
four decimal places.)
I am honestly so lost... if you could please show work I would
greatly appreciate it!!

Can someone solve these for me?
1.) Show that (F_(n-1))(F_(n+1)) - (F_n)^2 = (-1)^n for all n
greater than or equal to 1. (F_n is a fibbonaci sequence)
2.) Use induction to prove that 6|(n^3−n) for every integer n
≥0

use logarithmic differentiation to find the derivative of
y=\frac{(x+3)^2}{(x+2)^5(x+4)^7}

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