1. Limits and Continuity

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(x)=(2x−3)^2/3

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.

Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (-2,2) at which f is not continuous.

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

lim t → 0 sin (π/2 cos (tan t))

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (0, 3) at which f is not continuous.

Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.

Explain why the equation cos x = x has at least one solution.

Use the graph of $f\(\left\)(x\(\right\))$ to determine if the function is continuous or discontinuous at $x=c$.

$c=-2$

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>

Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.

Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0≤t≤60. <FIGURE>