0 lim ( ) = = indeterminate 0 x a if fx then say 1 lim () = undefined 0 x a if fx then say In any limit question, the first step is to replace x by the value given and check your answer as it may be: indeterminate, undefined, or any real number Case.1: Indeterminate you should i lif I f ti t t f t a very big number a very small number 0 simplify. In fractions you try to factor or 2 lim .. 2.000..1 x has x approaching 2 lim .. -1.999..9 x has x approaching lim .. - 1.999..9 has x approaching 3 3 7 . 7 . take common denominator then cancel like expressions. In radicals you conjugate or (rationalize). Case.2: Undefined you should apply the two sided does notimply differentiability 2 lim .. 1.999..9 x has x approaching 2 lim .. -2.000..1 x has x approaching Case.2: Undefined you should apply the two sided limits for it may turn out to be unequal and in this case you say the limit does not exist DNE Differentiable means continuous but continuity d ti l li () li () li () () f f f fb f(x) is a continuous function iff Rational functions: May admit Asymptotes Vertical (V) put q(x)=0 and find x=number lim () lim () lim () () x b x b x b f x f x f x fb If the relation is false the f(x) is a discontinuous function Vertical (V) put q(x)=0 and find x=number Horizontal (H) () () px y qx () lim () x p x find qx Check your answer. If indeterminate then simplify. You get y=number, or as no (H) Rate of change=average rateof change=y/ x get y number, or as no (H) = = dy/dx= slope of the tangent line to the curve = slope of the curve at the point of tangency = = f’(x) = y’=four‐step process 2 1 2 1 ( ) ( ) y y x x 0 ( ) () lim h f x h fh h Slope=y’= f’(x)=0 for a horizontal line =1/0 for a vertical line Implicit derivative: deriv. of y is y’ then deriv of y 2 is 2y y’ and deriv of 4y 3 is derivative of y w.r.t x = velocity = V if f(x) stands for the position of a moving object at time x h Differentials: y’=f’=dy/dx then dy=f’.dx So if Area=r 2 then dA=2 r.dr where dA represents the change in area and dr the change in r =increase or decrease of r if + ‐ Related Rates: including the factor of time relating variables to time derivative of deriv . of y is 2y . y and deriv . of 4y is 12y 2 y’ while that of x has no x’ for its value is 1. WHY? Because x is independent and y depends on x for y=f(x). So in any implicit Related Rates: including the factor of time relating variables to time derivative of y y’ dy/dx because no more we are relating y w.r.t x but in fact we are relating each w.r.t time ‘t’. So in this context derivative of y=change in rate of y=dy/dt. Same for dx/dt and deriv. of x 2 =2xdx/dt. y also known as increment in y. same for x= dx. While dy=df is the differential of y =part of the whole derivative y’ or f’. equation you derivate with the presence of y’, then collect y’. 2 2 ': (1. ') 2 ' 2 . xy xy find y e y x e y cont yx yy x