# Lim 3 exists lim 2 defined is 1 c f x f x f c f c x c

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) ( ) ( lim 3. exists ) ( lim 2. defined is ) ( 1. c f x f x f c f c x c x
CONTINUITY OF A FUNCTION If functions f and g are continuous functions at x = c, then the following are true: a. f + g is continuous at c b. f g is continuous at c c. fg is continuous at c d. f/g is continuous at c provided g( c ) is not zero.
The figure above illustrates that the function is discontinuous at x=c and violates the first condition. The figure above illustrates that the function is discontinuous at x = c and violates the second condition. This kind of discontinuity is called jump discontinuity. Types of Discontinuity
The figure above illustrates that the limit coming from the right and left of c are both undefined, thus the function is discontinuous at x = c and violates the second condition. This kind of discontinuity is called infinite discontinuity. The figure above shows that the function is defined at c and that the limit coming from the right and left of c both exist thus the two sided limit exist. However, Thus, the function is discontinuous at x = c, violating the third condition. This kind of discontinuity is called removable discontinuity ( missing point). f ( c ) ¹ lim x ® c f ( x )
CONTINUITY OF A FUNCTION Examples: 1. Let ? ? = 2?, 𝑖? ? ≤ 3 3? − 3, 𝑖? ? > 3 . Determine if f(x) is continuos at x=3. Thus, f(x) is continuous at x=3. 6 ) 3 ( ) ( lim c. . 6 ) ( lim , , 6 ) 3 ( 2 ) ( lim , 6 3 ) 3 ( 3 ) ( lim b. defined. is 6 ) 3 ( 2 ) 3 ( a. 3 3 3 3 f x f x f thus x f x f f x x x x
2. Investigate the discontinuity of the function f defined. What type of discontinuity is illustrated? a) c. b) Show the point(s) of discontinuity by sketching the graph of the function . f x ( 29 = 2 x 3 - x + 3 f ( x ) = x 2 - 4 x - 2 , x 2 4, x = 2 f x ( 29 = x 2 - 1 2 x + 4 x 2 - 1 2 x x < 2 2 x < 6 x 6
2. Find values of the constants k and m, if possible, that will make the function f(x) defined as be continuous everywhere. f ( x ) = x 2 + 5 m ( x + 1) + k 2 x 3 + x + 7 x 2 - 1 < x < 2 x ≤ - 1
THE DERIVATIVE OF A FUNCTION The process of finding the derivative of a function is called differentiation and the branch of calculus that deals with this process is called differential calculus . Differentiation is an important mathematical tool in physics, mechanics, economics and many other disciplines that involve change and motion.
y )) ( , ( 1 1 x f x P )) ( , ( 2 2 x f x Q ) ( x f y x x x x x x 1 2 1 2 tangent line secant line x Consider: - Two distinct points P and Q -Determine slope of the secant line PQ - Investigate how the slope changes as Q approaches P. - Determine the limit of the secant line as Q approaches P .
DEFINITION : Suppose that is in the domain of the function f, the tangent line to the curve at the point is with equation 1 x ) ( x f y )) ( , ( 1 1 x f x P ) ( ) ( 1 1 x x m x f y where provided the limit exists, and is the point of tangency. )) ( , ( 1 1 x f x P x x f x x f m x ) ( ) ( lim 1 1 0
DEFINITION The derivative of at point P on the curve is equal to the slope of the tangent line at P, thus the derivative of the function f with respect to x, given by , at any x in its domain is defined as: ) ( x f y 0 0 ( ) ( ) lim lim x x dy y f x x f x dx x x      