1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at...
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Transcript of 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at...
![Page 1: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/1.jpg)
1.3 – Continuity, End Behavior, and Limits
![Page 2: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/2.jpg)
![Page 3: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/3.jpg)
![Page 4: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/4.jpg)
Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.
a. f(x) = 3x – 2 if x > -3 ; at x = -3 2 – x if x < - 3
![Page 5: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/5.jpg)
Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.
a. f(x) = 3x – 2 if x > -3 ; at x = -3 2 – x if x < - 3
1. Find f(-3).
![Page 6: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/6.jpg)
Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.
a. f(x) = 3x – 2 if x > -3 ; at x = -3 2 – x if x < - 3
1. Find f(-3). f(-3) = 2 – (-3) = 5
![Page 7: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/7.jpg)
Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.
a. f(x) = 3x – 2 if x > -3 ; at x = -3 2 – x if x < - 3
1. Find f(-3). f(-3) = 2 – (-3) = 5, so f(-3) exists
![Page 8: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/8.jpg)
2. Investigate values close to f(-3)
![Page 9: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/9.jpg)
2. Investigate values close to f(-3)
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) 5.1 5.01 5.001 -10.997 -10.97 -10.7
![Page 10: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/10.jpg)
2. Investigate values close to f(-3)
As x -3 from left, f(x) 5
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) 5.1 5.01 5.001 -10.997 -10.97 -10.7
![Page 11: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/11.jpg)
2. Investigate values close to f(-3)
As x -3 from left, f(x) 5As x -3 from right, f(x) -11
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) 5.1 5.01 5.001 -10.997 -10.97 -10.7
![Page 12: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/12.jpg)
2. Investigate values close to f(-3)
As x -3 from left, f(x) 5As x -3 from right, f(x) -11Since don’t approach same value,
discontinuous and jump discontinuity.
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) 5.1 5.01 5.001 -10.997 -10.97 -10.7
![Page 13: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/13.jpg)
b. f(x) = x + 3 ; at x = -3 and x = 3 x2 – 9
![Page 14: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/14.jpg)
b. f(x) = x + 3 ; at x = -3 and x = 3 x2 – 9
1. Find f(-3) and f(3).
![Page 15: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/15.jpg)
b. f(x) = x + 3 ; at x = -3 and x = 3 x2 – 9
1. Find f(-3) and f(3).f(-3) = -3 + 3 = 0 = Ø
(-3)2 – 9 0
![Page 16: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/16.jpg)
b. f(x) = x + 3 ; at x = -3 and x = 3 x2 – 9
1. Find f(-3) and f(3).f(-3) = -3 + 3 = 0 = Ø
(-3)2 – 9 0f(3) = 3 + 3 = 6 = Ø (3)2 – 9 0
![Page 17: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/17.jpg)
b. f(x) = x + 3 ; at x = -3 and x = 3 x2 – 9
1. Find f(-3) and f(3).f(-3) = -3 + 3 = 0 = Ø
(-3)2 – 9 0f(3) = 3 + 3 = 6 = Ø (3)2 – 9 0
Since both f(-3) = Ø and f(3) = Ø, f(x) is discontinuous at both x = -3 and x = 3.
![Page 18: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/18.jpg)
2. Investigate values close to f(-3) and f(3).
![Page 19: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/19.jpg)
2. Investigate values close to f(-3) and f(3).
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
![Page 20: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/20.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
![Page 21: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/21.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
![Page 22: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/22.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
Since they approach same value, limit exists.
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
![Page 23: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/23.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
Since they approach same value, limit exists.
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
x 2.9 2.99 2.999 3.0 3.001 3.01 3.1
f(x) -10 -100 -1000 1000 100 10
![Page 24: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/24.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
Since they approach same value, limit exists.
As x 3 from left, f(x) -∞
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
x 2.9 2.99 2.999 3.0 3.001 3.01 3.1
f(x) -10 -100 -1000 1000 100 10
![Page 25: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/25.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
Since they approach same value, limit exists.
As x 3 from left, f(x) -∞As x 3 from right, f(x) ∞
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
x 2.9 2.99 2.999 3.0 3.001 3.01 3.1
f(x) -10 -100 -1000 1000 100 10
![Page 26: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/26.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
Since they approach same value, limit exists.
As x 3 from left, f(x) -∞As x 3 from right, f(x) ∞
Since limit x -3 exists but f(-3) doesn’t, removable discontinuity.
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
x 2.9 2.99 2.999 3.0 3.001 3.01 3.1
f(x) -10 -100 -1000 1000 100 10
![Page 27: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/27.jpg)
2. Investigate values close to f(-3) and f(3).
As x -3 from left, f(x) -0.167As x -3 from right, f(x) -0.167
Since they approach same value, limit exists.
As x 3 from left, f(x) -∞As x 3 from right, f(x) ∞
Since limit x -3 exists but f(-3) doesn’t, removable discontinuity.Since limit x -3 doesn’t exist, infinite discontinuity.
x -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9
f(x) -0.164 -0.166 -0.167 -0.167 -0.167 -0.169
x 2.9 2.99 2.999 3.0 3.001 3.01 3.1
f(x) -10 -100 -1000 1000 100 10
![Page 28: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/28.jpg)
Ex. 2 Use the graph of the function to describe its end behavior.
![Page 29: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/29.jpg)
Ex. 2 Use the graph of the function to describe its end behavior.
lim f(x) = - ∞ x - ∞
![Page 30: 1.3 Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.](https://reader034.fdocuments.net/reader034/viewer/2022050807/5a4d1b6e7f8b9ab0599b44df/html5/thumbnails/30.jpg)
Ex. 2 Use the graph of the function to describe its end behavior.
lim f(x) = - ∞ x - ∞
lim f(x) = - ∞ x - ∞