In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
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In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
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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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b. Does lim x → −1⁺ f (x) exist?
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?
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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At what values of x is f continuous?
At what points are the functions in Exercises 13–30 continuous?
y = 1/(x – 2) – 3x
At what points are the functions in Exercises 13–30 continuous?
y = 1/(|x| + 1) − x²/2
At what points are the functions in Exercises 13–30 continuous?
y = √(x⁴ +1)/(1 + sin² x)
At what points are the functions in Exercises 13–30 continuous?
y = (2x – 1)¹/³
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 t → 0 sin (π/2 cos (tan t))
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim x → 0 tan (π/4 cos (sin x¹/³))
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)
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim x → 0 sin ((π + tan x)/(tan x – 2 sec x))
Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim ϴ → 0 cos (πϴ/sin ϴ)
Define h(2) in a way that extends h(t) = (t² + 3t − 10)/(t − 2) to be continuous at t = 2.