1. Limits and Continuity

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)

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.

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

y = √(2x + 3)

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

g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3

What does it mean for a function to be continuous on an interval?

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

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

y = 1/(x – 2) – 3x

Limits and Continuity

Repeat the instructions of Exercise 1 for

1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Limits and Continuity

On what intervals are the following functions continuous?

b. g(x) = csc x

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

f(x) = { (x³ − 8)/(x² − 4), x ≠ 2, x ≠ −2

3, x = 2

4, x = −2

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → π/6 √(csc² x + 5√3 tan x)

Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 

f(x)=1+sin x / cos x; limx→π/2^− f(x); lim x→4π/3 f(x)

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={2x if x<1

x^2+3x if x≥1; a=1

Evaluate f(3) if lim x→3^− f(x)=5,lim x→3^+ f(x)=6, and f is right-continuous at x=3.