Section 2 5 Limits involving infinity I Infinity Minus Infinity as a limit as x approaches a The line x a is a vertical asymptote of the function Basic limits to remember lim x 0 1 x Find each limit for the function f x 1 lim x 0 x x 2 3 x 10 lim ln x x 0 x2 4 a lim f x This is a rational function and substitution gives the indeterminate form 0 0 x 2 x 5 x 2 We factor numerator and denominator x 5 b lim f x lim x 2 x 2 x 2 x 2 x 5 x 2 7 4 as x 2 DNE because substitution gives 3 0 x 2 is a vertical x 2 asymptote In fact the right side approaches positive infinity the left side approaches minus infinity We can check the sign of the expression for x 2 and see it is check the sign for x 2 and see it is negative or follow c and d below c lim x 2 f x d Similarly We write lim x 0 x 2 lim x 2 1 x2 x 5 lim x 2 lim x 2 3 x 2 0 x 2 lim t 0 t 3 t f x since both sides approach positive infinity Example Find lim x 0 ln x x or show it does not exist ln x is approaching minus infinity on both sides of 0 x is changing sign and is negative on the left positive on the right lim x 0 ln x x lim x 0 ln x x and lim x 0 ln x x does not exist II Limits as x approaches positive infinity and as x approaches minus infinity The tendency of a polynomial as x approaches plus or minus infinity is the same as the tendency of its leading term You can see this two ways Graph Y 1 x 3 5 x 2 3 x 10 Y 2 x 3 in your calculator Use a window in which Xmin and Xmax are large say Xmin 1000 and Xmax 1100 The graphs are nearly identical If x 1000 then the leading term is on the order of 1000 times any other term lim x 3 lim Y 1 so x lim x 3 so x The same works at minus infinity x lim Y 1 x Ex lim 2 x 5 20 x 4 15 x 3 35 lim 2 x 5 x x Why is this Factor out the leading term to get 2 x 5 1 10 x 15 2x 2 35 2x5 The term in parentheses approaches 1 as x approaches infinity or infinity Any odd power of x will approach minus infinity as x approaches minus infinity because an odd power of a negative is negative Then the coefficient 2 makes it positive Rational functions You only need to consider the tendency of the ratio of the leading terms Example lim 2 x 4 15 x 3 16 x 2 x Example lim x 6 x 4 32 x 3 50 x lim x 3 x 2 4 x 10 2 x 3 5 x 2 10 x 15 2x4 6x4 lim x 3x 2 2x3 2 6 1 3 lim x y 1 3 is a horizontal asymptote 3 2x 0 asymptote Example lim x 4 x 4 3x 2 9 3 5x 2 x 2 lim x 4x4 5x 3 lim x 4x 5 y 0 is a horizontal Other limits infinity minus infinity is indeterminate Example Find lim x 2 4 x x 2 1 Both terms tend to infinity so we have x infinity minus infinity The limit could be anything or not exist We must do algebra x2 4x x2 1 4x 1 x2 4x x2 1 x2 4x x 2 1 x 2 4 x x2 4x 4x 2x 2 x 2 4 x x 2 1 x2 4x x2 1 The means has the same limit as Exercises Show lim x 2 6 x x 3 and x x2 1 x 2 1 lim x 2 6 x x 3 x Show that every odd degree polynomial has at least one root Use the Intermediate Value Theorem
View Full Document
Unlocking...