1. Limits and Continuity

Limits and Continuity

On what intervals are the following functions continuous?

d. k(x) = sin x / x

Which of the following functions are continuous for all values in their domain? Justify your answers.

d. p(t)=number of points scored by a basketball player after t minutes of a basketball game

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

lim ϴ → 0 cos (πϴ/sin ϴ)

Determine the interval(s) on which the following functions are continuous. 

f(x)=1 / x^2−4

Limits and Continuity

On what intervals are the following functions continuous?

a. ƒ(x) = x¹/³

Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = cos x / x―π

Limits and Continuity

On what intervals are the following functions continuous?

b. g(x) = x³/⁴

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.

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$g\left(x\right)=\cos \left(e^x\right)$

A function value Show that the function F(x) = ( x − a)²(x − b)² + x takes on the value (a + b)² for some value of x.

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

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a. Does f (1) exist?

Limits of quotients

Find the limits in Exercises 23–42.

limu→1 (u⁴ − 1)/(u³ − 1)

Suppose f(x) = {x^2 − 5x + 6 / x − 3 if x≠3

a if x=3.

Determine a value of the constant a for which lim x→3 f(x) = f(3).

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

y = √(x⁴ +1)/(1 + sin² x)

Which of the following functions are continuous for all values in their domain? Justify your answers.

c. T(t)=temperature t minutes after midnight in Chicago on January 1