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Section 2.8 - Continuity
2.2
A function f(x) is continuous at x = c if and only if all three of the following tests hold:
i. f c exists
x c
ii. lim f x exists
x c
iii. lim f x f c
f(x) is right continuous at x = -5
f(x) is continuous at x = -4
f(x) has infinite discontinuity at x = -3 [i, iii]
f(x) has point discontinuity at x = -2 [i, iii]
f(x) has infinite discontinuity at x = -1 [i, ii, iii]
f(x) is continuous at x = 0
At x = 1
At x = 2
At x = 3
At x = 4
At x = 5
Point Discontinuity [i, iii]
Jump Discontinuity [i, ii, iii]
Continuous
Continuous
Point Discontinuity [i, (ii), iii]
2f x x 2x 1
continuous
2
1f xx 1
continuous
2
xf xx x
xx x 1
pt. discontinuity at x = 0inf. discontinuity at x = 1
2
x 3f xx 9
x 3
x 3 x 3
pt. discontinuity at x = 3inf. discontinuity at x = -3
2
2x 3 x 1f x
x x 1
2 1 3 1
21 1
continuous
1 x 1 x 2
f x 23 x x 2
1 2 1 22
3 2 1
jump discontinuity at x = 2
Find the value of a which makes the function below continuous
3
2
x x 2f x
ax x 2
3
2
3
2
x x 2f x
ax x
2 8
2 4a2 a
4a 8 a 2
Find (a, b) which makes the function below continuous
2 x 1
f x ax b 1 x 32 x 3
As we approach x = -1 2 = -a + b
As we approach x = 3 -2 = 3a + b
4 4aa 1 b 1
1,1
Consider the function sinx x 0
f x xk x 0
Find the value of k which makes f(x) continuous at x = 0
Sincex 0
sinxlim 1x
, if k =1, the hole is filled.
Calculator RequiredLet m and b be real numbers and let the function f be defined by:
21 3bx 2x x 1
f xmx b x 1
If f is both continuous and differentiable at x = 1, then:A. m 1, b 1B. m 1, b 1C. m 1, b 1D. m 1, b 1E. none of these
3b 4x x 1f ' x
m x 1
3b 4 m
1 3b 2 m b
3b m 42b m 3
No CalculatorThe function f is continuous at x = 1
x 3 3x 1 x 1If f x then kx 1
k x 1
1 1A. 0 B.1 C. D. E. none of these2 2
xx 3 3x 1 3 3x 1xx 31 3
f xx 1
x 3 3x 1
3 3x 1x x1
x 1
2 x 1
x 3 3x 1
Calculator Required
Which of the following is true about 2
2
x 1f x ?
2x 5x 3
I. f is continuous at x = 1 II. The graph of f has a vertical asymptote at x = 1III. The graph of f has a horizontal asymptote at y = 1/2
A. I B. II C. III D. II, III E. I, II, III
I. f(1) results in zero in denominator….NO
II. Since x – 1 results in 0/0, it is a HOLE, NOT asymptote
III. 2
2x
x 1lim
2x 5x 3
2
2x
x 2x 1lim2x 5x 3
X XX X
12
No CalculatorWhich function is NOT continuous everywhere?
2 / 3
2
2
2
A. y x
B. y x
C. y x 1xD. y
x 14xE. y
x 1
undefined at x = -1
The graph of the derivative of a function f is shown below.Which of the following is true about the function f? I. f is increasing on the interval (-2, 1) II. f is continuous at x = 0III. f has an inflection point at x = 2
A. I B. II C. III D. II, III E. I, II, III
NOYESNO
Calculator Required
tanxConsider the function f defined on x by f x2 2 sinx
No Calculator
for all x . If f is continuous at x = , then f
A. 2 B. 1 C. 0 D. -1 E. -2
sinx
tanx cos xf xsinx sinx
sinx 1cos x sinx
1 1 1 Dcosx cos