Please answer these with lots of detail. I want to be able to understand the topic and material well.

Step-by-step explanation:

Q3

(a) If 0 < |x − 1| < δ, then |f(x) − 2| < 2

What this means is, how far can x stray from the x=1 line such that f(x) stays within 2 units of the y=2 line (0 < f(x) < 4).

If we move 2 units left of x=1, we get f(x) = 4.

If we move about 3.5 units right of x=1, we get f(x) = 4.

Therefore, δ can't be more than 2.

(b) If 0 < |x| < δ, then |f(x) − 3| < 1

What this means is, how far can x stray from the x=0 line such that f(x) stays within 1 unit of the y=3 line (2 < f(x) < 4).

If we move 1 unit left of x=0, we get f(x) = 4.

If we move 1 unts right of x=0, we get f(x) = 2.

Therefore, δ can't be more than 1.

Since f(x) isn't continuous within this domain, we can't conclude that the limit exists.

Q4

(a) Yes.  If δ = 0.25, then 0.75 < x < 1.25, and f(x) > 200.

(b) No.  f(1) = 300, so even if δ = 0, f(x) will be less than 400.

(c) Yes.  If δ ≈ 0.1, then 0.9 < x < 1, and f(x) > 450.

the correct answer option is (2x-3)(2x+3).

step-by-step explanation:

we are given the following expression and we are to factorize it:

we are to find such factors that when multiplied they give a result of 36 (9 x 4 = 36) and when added they give a result of 0.

the two such factors of 36 are negative six and postive six.

factorizing the given expression to get:

therefore, the correct factorization of is (2x-3)(2x+3).

answer: cos [   arcsin (3/5) - arccos (1/2) ] =

step-by-step explanation: × = 3/5 = 0.6--> x = 36.87 and x = 180 - 36.87 = 143.13 deg

cos y = 1/2 --> y = +- 60 deg a. cos (x - y) = cos (143.13 - 60) = cos 83,13  

=

0.12

b. cos (36.87 - 60) = cos (-23.23) = 0.92