Question

Let f be defined on the (0,infinity). Prove that the limit as x approaches infinity of F(x) =L if and only if the limit as x approaches 0 from the right of f(1/x) = L. Does this hold if we replace L with either infinity or negative infinity?

Answer #1

Prove that a sequence (un such that n>=1)
absolutely converges if the limit as n approaches infinity of
n2un=L>0

Given a metric space Y, a point L in Y, and f:[0,infinity) ->
Y, f has a limit L element of Y at infinity, written, if for every
epsilon greater than 0 there is a C>0 such that if x>C then
d(f(x),L) < epsilon. Prove, if f:[0,infinity) -> Y is
continuous and has a limit at infinity, then f is uniformly
continuous.

Estimate the lim as f(x) approaches -infinity by graphing
f(x)=sqrt{x^{2}+x+9}+x
(b) Use a table of values of f(x) to guess the
value of the limit. (Round your answer to one decimal place.)
(c) Prove that your guess is correct by evaluating
lim x→−∞ f(x).

Let f : [0,∞) → [0,∞) be defined by, f(x) := √ x for all x ∈
[0,∞), g : [0,∞) → R be defined by, g(x) := √ x for all x ∈ [0,∞)
and h : [0,∞) → [0,∞) be defined by h(x) := x 2 for each x ∈ [0,∞).
For each of the following (i) state whether the function is defined
- if it is then; (ii) state its domain; (iii) state its codomain;
(iv) state...

Let f:[0,1]——>R be define by f(x)= x if x belong to rational
number and 0 if x belong to irrational number and let g(x)=x
(a) prove that for all partitions P of [0,1],we have
U(f,P)=U(g,P).what does mean about U(f) and U(g)?
(b)prove that U(g) greater than or equal 0.25
(c) prove that L(f)=0
(d) what does this tell us about the integrability of f ?

sketch a neat, piecewise function with the following
instruction: 1. as x approach infinity, the limit of the function
approaches an integer other than zero. 2. as x approaches a
positive integer, the limit of the function does not exist. 3. as x
approaches a negative integer, the limit of the function exists. 4.
Must include one horizontal asymtote and one vertical asymtote.

Let f: [0, 1] --> R be defined by f(x) := x. Show that f is
in Riemann integration interval [0, 1] and compute the integral
from 0 to 1 of the function f using both the definition of the
integral and Riemann (Darboux) sums.

Given a function f:R→R and real numbers a and L, we say that
the limit of f as x approaches a is L if for all ε>0, there
exists δ>0 such that for all x, if 0<|x-a|<δ, then
|f(x)-L|<ε. Prove that if f(x)=3x+4, then the limit of f as x
approaches -1 is 1.

the global maximum of the function h(x)=1/x^2 + 1 defined on
(-infinity, infinity) is: a) 1 b) 0 c) 1/2 d) DNE
Consider a family of functions f(x)=e^-ax + e^ax for a does not
= 0. which of the following holds for every member of the
family?
a) f is always increasing b) f is always concave up c) f has no
critical points

(a) Show that the function f(x)=x^x is increasing on (e^(-1),
infinity)
(b) Let f(x) be as in part (a). If g is the inverse function to
f, i.e. f and g satisfy the relation x=g(y) if y=f(x). Find the
limit lim(y-->infinity) {g(y)ln(ln(y))} / ln(y). (Hint :
L'Hopital's rule)

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