Limits Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 18 - PowerPoint PPT Presentation

Limits Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 18

Section 3.1 :: Limits 2 / 18

Informal Definition of Limit We write x → a f ( x ) = L lim and say “the limit of f ( x ), as x approaches a , equals L ” if we can make the values of f ( x ) arbitrarily close to L (as close to L as we like) by taking x sufficiently close to a (on either side of a ), but not equal to a . 3 / 18

Find the Value of the Limit x 2 − 3 x + 1 � � lim x → 2 4 / 18

Find the Value of the Limit � x 2 − x − 1 � lim √ x + 1 x → 3 5 / 18

Find the Value of the Limit � x 2 − x − 6 � lim x + 2 x →− 2 6 / 18

Find the Value of the Limit � √ x − 5 � lim x − 25 x → 25 7 / 18

One-Sided Limits We write x → a − f ( x ) = L lim and say the left-hand limit of f ( x ) as x approaches a [or the limit of f ( x ) as x approaches a from the left] is equal to L if we can make the values of f ( x ) arbitrarily close to L by taking x to be sufficiently close to a with x less than a . 8 / 18

One-Sided Limits We write x → a + f ( x ) = L lim and say the right-hand limit of f ( x ) as x approaches a [or the limit of f ( x ) as x approaches a from the right] is equal to L if we can make the values of f ( x ) arbitrarily close to L by taking x to be sufficiently close to a with x greater than a . 9 / 18

Splitting Limits by Sides x → a f ( x ) = L lim if and only if x → a − f ( x ) = L and lim x → a + f ( x ) = L lim 10 / 18

Informal Definition of Limit at Infinity Let f be a function defined on some interval ( a , ∞ ) . Then x →∞ f ( x ) = L lim means that the values of f ( x ) can be made arbitrarily close to L by taking x sufficiently large. 11 / 18

Horizontal Asymptotes The line y = L is called a horizontal asymptote of the curve y = f ( x ) if either x →∞ f ( x ) = L lim or x →−∞ f ( x ) = L lim 12 / 18

Determine the Limit at Infinity � 5 x − 1 � lim 2 x + 2 x →∞ 13 / 18

Determine the Limit at Infinity � 15 x 2 − 7 x − 1 � lim 3 x 2 + 100 x →∞ 14 / 18

Determine the Limit at Infinity � x − 1 � lim 3 x 2 + 2 x →∞ 15 / 18

Determine the Limit at Infinity � x 2 − x � lim 5 x + 4 x →∞ 16 / 18

Average Cost The cost (in dollars) for manufacturing a particular DVD is C ( x ) = 15 , 000 + 6 x , where x is the number of DVDs produced. Recall that the average cost per DVD, denoted by C ( x ), is found by dividing C ( x ) by x . Find and interpret lim x →∞ C ( x ). 17 / 18

Sediment To develop strategies to manage water quality in polluted lakes, biologists must determine the depths of sediments and the rate of sedimentation. It has been determined that the depth of sediment D ( t ) (in centimeters) with respect to time t (in years before 1990) for Lake Coeur d’Alene, Idaho, can be estimated by the equation D ( t ) = 155(1 − e − 0 . 0133 t ) , Source: Mathematics Teacher. (a) Find D (20) and interpret. (b) Find lim t →∞ D ( t ) and interpret. 18 / 18