MATH 1225 Lecture Notes - Lecture 4: Trigonometric Functions, Inverse Trigonometric Functions, Function Composition

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28 May 2018

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Sec. 2.5 Continuity

NOTE: This section deviates from the book’s terminology

• A curve is continuous if it has no holes, breaks, or vertical asymptotes

• Definition of a Continuous Function: A function

is continuous at a number

   

afxf

ax 



lim

o Note: 3 things are required if

is continuous at

▪

 

is defined (

is in the domain of

)

▪

 

ax

lim

exists

▪

   

afxf

ax 



lim

• Definition of a Discontinuous Function: A function

is discontinuous at

ax 

is not continuous at

ax 

is in the domain of

EX: Why are the graphs below not continuous based on the definition of a continuous function? Which of these

graphs admits a continuous extension or correction at

ax 

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Document Summary

Note: this section deviates from the book"s terminology: a curve is continuous if it has no holes, breaks, or vertical asymptotes, definition of a continuous function: a function f is continuous at a number a if xf. Af lim x a: note: 3 things are required if f is continuous at a. Af lim x a lim x a is defined ( a is in the domain of f ) Af: definition of a discontinuous function: a function f is discontinuous at x if a ax f is not continuous at ax is in the domain of f. Which of these graphs admits a continuous extension or correction at ax : definition: a function f is continuous from the right at a number a if xf. Af lim x a and function f is continuous from the left at a if xf. 0: is f continuous from the left at.

Related Questions

4. (21 pts) Let g be the function given by ( 6 - X 2 + x if x < 2, x + -2 g(x) = x2 – x-1 if x>2. (a) (6 pts) Use the definition of continuity to determine whether the function g is continuous at 2. Show your work. (b) (3 pts) Evaluate the limit. Justify your answer. Show your work. Write "does not exist" only if the limit does not exist and is not + or -0. lim g(x) = lim -2+ = +00 X 2 + X since the form of the limit is NUM: POS, DENOM: POS (c) (2 pts) Write the equation(s) of all vertical asymptotes of g. Write "none" if appropriate. The result in part (b) implies that the line x= -2 is a vertical asymptote. This is the only vertical asymptote, since the function g is continuous on its domain (d) (2 pts) Determine the largest) intervals of continuity of g. Use interval notation to write your answer. Let g be the function given by 12+ if x < 2, x# -2 2 + x g(x)= x2-x-1 x-1 if x>2. (e) (6 pts) Evaluate each limit. Do not use L'Hôpital's Rule. Write "does not exist" only if the limit does not exist and is not too or -00. Show your work. 6-1 i lim g(x) = lim -00 2 + X = lim -002 + x 0-1 + 1 x -00 x x x -+- 2 +1 0 0 + 11 x2 - x-1 1 ii. lim g(x) = lim x++ V x2 – x-1 X-1 x2 - x-1 X - 1 = lim x +oc x +0 - lim Vam M93001 (f) (2 pts) Write the equation(s) of all horizontal asymptotes of g. Write "none" if appropriate.

1. (18 pts) The graph of a function f, defined on (-0, +o), is given in the figure below. - - y = f(x) - - - - - - N - - 4 A -2-1 144 -1 - - - - - - - -- - - -- Use the graph of f to answer questions (a)-(f) below. (a) Determine the value or write "does not exist". i. f(-3) = -2 ii. f(0) = 1 (b) Determine the limit or write "does not exist" (DNE). Write "does not exist" only if a limit does not exist and is not too or -0. i. lim f(x) = 0 iv. lim f(x) = +00 X- 3 x2 ii. lim f(x) = -4 v. lim f(x) = -0 X- 0- x -00 iii. lim f(x) = DNE vi. lim f(x) = -2 x0 x++00 (C) MULTIPLE CHOICE! Circle ONLY one answer! Find a number a for which the following statement is true (the function f is shown above): STATEMENT: The function f is continuous at a, and lim f(x) = 0. i. a=-3 iii. a=0 V. a=3 ii. a = -2 iv. a = 2 vi. Such a number does not exist. (d) Determine the largest intervals of continuity of f. Use interval notation to write your answer. (-0, -3), (-3,0), (0, 2), (2, +00) (e) Write the equation(s) of all vertical asymptotes of f. Write "none" if appropriate. x= 2 (f) Write the equation(s) of all horizontal asymptotes of f. Write "none" if appropriate. y = -2

MATH 1225 Lecture Notes - Lecture 4: Trigonometric Functions, Inverse Trigonometric Functions, Function Composition

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