Question

4. Please work each part. (a) Discuss the existence or non-existence of limx→0 2 sin 1 x − x 2 cos 1 x using the limit theorems. (b) Let I be an open interval with a ∈ I and suppose that f is a function defined on I\{a}. Suppose that limx→a (f(x) + D(x)) exists, where D(x) = χQ(x) is the Dirichlet function. Show that limx→a f(x) does not exist.

Answer #1

5. Let I be an open interval with a ∈ I and suppose that f is a
function defined on I\{a} where the limit of f exists as x → a and
L = limx→a f(x). Prove that the limit of |f| exists as x → a and
|L| = limx→a |f(x)|. Is the converse true? Prove or furnish a
counterexample.

use
the squeeze theorum to show that
*** please show work
limx→0 cos(x)x^8 sin(1/x)=0
limx→0 tan(x)x^4 cos(2/x)=0

4. Let f be a function with domain R. Is each of the following
claims true or false? If it is false, show it with a
counterexample. If it is true, prove it directly from the FORMAL
DEFINITION of a limit.
(a) IF limx→∞ f(x) = ∞ THEN limx→∞ sin (f(x)) does
not exist.
(b) IF f(−1) = 0 and f(1) = 2 THEN limx→∞ f(sin(x)) does not
exist.

If f is a continuous, positive function defined on the interval
(0, 1] such that limx→0+ = ∞ we have seen how to make sense of the
area of the infinite region bounded by the graph of f, the x-axis
and the vertical lines x = 0 and x = 1 with the definition of the
improper integral.
Consider the function f(x) = x sin(1/x) defined on (0, 1] and
note that f is not defined at 0.
• Would...

Calculus, Taylor series Consider the function f(x) = sin(x) x .
1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s
remainder theorem to get the same result: (a) Write down P1(x), the
first-order Taylor polynomial for sin(x) centered at a = 0. (b)
Write down an upper bound on the absolute value of the remainder
R1(x) = sin(x) − P1(x), using your knowledge about the derivatives
of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x...

Consider the function on the interval (0, 2π).
f(x) =
sin(x)/
2 + (cos(x))2
(a) Find the open intervals on which the function is increasing
or decreasing. (Enter your answers using interval notation.)
increasing
decreasing
(b) Apply the First Derivative Test to identify the relative
extrema.
relative maximum
(x, y) =
relative minimum
(x, y) =

Evaluate each limit. Use l' Hospital's Rule if appropriate.
a) limx→−3 81−x^4/9x+27
b) limx→0 e^3x −1−3x/x^2
c) limx→0+ xlnx

find the derivative of each of the following show all work and
circle answer please
part 1) f(x)= x+2/x^2+2
part 2) f(x)=2tan(x)+sin(3x)-10
part 3) f(x)=x^2 . cos(x^3-2)
part 4) f(x)=ln((x^7 sqrt(x^3+1)/(3x^2+8)^5))

1) find the
absolute extrema of function f(x) = 2 sin x + cos 2x on the
interval [0, 2pi]
2)
is f(x) = tanx
concave up or concave down at x = phi / 6

double integral f(x,y)dA. f(x,y) = cos(x) +sin(y). D is the
triangle with vertices (-1,-2), (1,0) and (-1,2). You might notice
cos(x) is an even function, sin(y) is an odd function.

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