Question

Given that lim x→π/2 tan2(x) = [infinity], illustrate the definition by finding values of δ that correspond to the following. (Round your answer down to four decimal places.)Given that lim x→π/2 tan2(x) = [infinity], illustrate the definition by finding values of δ that correspond to the following. a. M = 500 δ=__________ b. M = 1,000 δ=________c. Find the largest number δ such that if |x − 1| < δ, then |4x − 4| < ε, where ε = 1.δ ≤ _________ d. Repeat and determine δ with ε = 0.1.δ ≤ _________e. Evaluate the piecewise defined function at the indicated values.f(x) =_______

Answer 1

Answer:

a) and b) δ = 0.01 works for both M= 500 and M=1000

c) We take δ = 1/4 = 0.25

d) we take δ = 1/40 = 0.025

Step-by-step explanation:

Approximately, $\frac{\pi}{2} = 1.57079632 .$ Lets evaluate tan²(x) in a truncated expression of this approximation

tan²(1) = 1.557² = 2.4255

tan²(1.5) = 14,1² = 198.85

tan²(1.57) = 1255² = 1255² = 1576947

Notice that this value clearly works for the 2 given values of M.

Lets work with $\delta = 0.01$ , since it didnt work for 0.1

tan²( π/2-0.01) = 10000 >1000>500

tan²(π/2+0.01) = 10000 (tan is asymmetrical respect to π/2, but tan² therefore is symmetrical).

δ = 0.01 works for both cases.

c) Note that if |x-1| < δ, then |4x-4| = |4(x-1)| = 4|x-1| < 4δ, for any value ε>0, we can take δ = ε/4.

if ε = 1, then we take δ = 1/4.

d) If ε = 0.1, then we take δ = 0.1 /4 = 0.025

e) f(x) = 4x

for ε = 1, we may take x = 0.75 and x = 1.25

Note that f(1) = 4

f(0.75) = 3

|3-4| = 1

f(1.25) = 5

|5-4| = 1

For ε = 0.1, we take x = 0.975 and x = 1.025

f(0.975) = 0.975*4 = 3.9

f(1.025) = 4*1.025 = 4.1

In both cases, the difference is 0.1. Also, if you take values closer from 1, the difference will be smaller.