At what points are the functions in Exercises 13–30 continuous?


y = √(2x + 3)

The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 3x − 1 = 0

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 15x + 1 = 0 (three roots)

At what points are the functions in Exercises 13–30 continuous?

y = cos (x) / x

At what points are the functions in Exercises 13–30 continuous?

g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3

Limits of quotients

Find the limits in Exercises 23–42.

limu→1 (u⁴ − 1)/(u³ − 1)

Is there a value of c that will make

f(x) = { (sin²(3x)) / x², x ≠ 0

c, x = 0

continuous at x = 0? Give reasons for your answer.

Limits and Continuity

Graph the function

1 , x ≤ ―1

―x , ―1 < x < 0

ƒ(x) = { 1 , x = 0 ,

―x , 0 < x < 1

1 , x ≥ 1

Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.