Continuous Extension


Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim x→(−π/2)⁺ sec x

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

b. Solve the equation ƒ(𝓍) = 0 graphically with an error of magnitude at most 10⁻⁸ .

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim θ→0 (2 − cot θ)

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

c. It can be shown that the exact value of the solution in part (b) is

(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³

Evaluate this exact answer and compare it with the value you found in part (b).

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2