Question

Evaluate each of the following limits. a) limx→∞(x/x − 2)^x

b) limx→0 sin x cos x/x + tan x

Answer #1

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. 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.

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 sin(x/2), cos(x/2), and tan(x/2) from the given
information.
Sin(x) = 24/25, 0° < x < 90°

Use the Maclaurin series for cos(?) , ????(?), and ??
to evaluate the following limits:
a. lim ?→0 −? − 1+?x / 5? 2 .
b. lim ?→0 ? − ???? / ?3 ????

solve the following: A) cos x = -(1/2) on [0, 2pi] B) sin x =
(-sqrt 3)/2 on [0, 2pi].

(a) Rewrite the expression as an algebraic expression in x
tan(sin-1(x))
(b) Find sin(2x), cos(2x) and tan (2x) from the given
information
csc(x)=4, tan(x)<0

Find the exact value of the expressions cos(a+b), sin(a+b) and
tan(a+b) under the following conditions:
cos(x)=12/13,x lies in quadrant 4, and sin(y)=-4/11, y lies in
quadrant 3
a. cos (a+b) b.sin(a+b) c.tan(a+b)

Use L'Hopitals Rule to evaluate the following limit.
lim x-> 0 ( sin(8x) - 8x cos(8x) ) / ( 8x - sin(8x) )
Please use L'Hopitals Rule please

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...

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