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CSC Worthy Warm-up Evaluate. Use your calculator and round to the nearest hundredth. 1. sin32.5 = cos _______ = .54 57.5 2. csc12.06 = sec _______ 77.94 = 4.79 3. log6127 = 2.70 Warm-up ( ) 1. sin 5 3 = − 3 2 3. cos 5. 2 −1 ( 2log 2 8 − 2 = 64 ) ( ) 2. tan − 3 4 = 1 −1 3 = 4 4. tan ( 0 ) = 0 2 6. e −2ln 7 = 149 7. 3§4.2¨ − 4§−2.5¨ + 2§−5¨ = 14 2 1 − x −1 8. tan ( cos x ) = x CSC Worthy Warm-up Sketch the graph of a function that satisfies all of the following properties. f (0) = −1, f (3) = 1,lim f ( x ) = 1,lim f ( x ) = − 2,lim f ( x ) = 2 x →0 x→3− x→3+ Warm-up CSC Worthy Section 2.1:5 If a ball is thrown into the air with a velocity of 40ft/s, the height in feet t seconds later is given by y = 40t – 16t2. a) Find the average velocity for the time period beginning when t = 2 and lasting . . . (i). O.5 seconds (ii). 0.1 seconds (iii). 0.05 seconds (iv). 0.01 seconds b) Estimate the instantaneous velocity when t = 2. Warm-up f ( x + h) − f ( x ) m= h CSC Worthy Section 2.1:5 If a ball is thrown into the air with a velocity of 40ft/s, the height in feet t seconds later is given by y = 40t – 16t2. a) Find the average velocity for the time period beginning when t = 2 and lasting . . . (i). O.5 seconds (ii). 0.1 seconds y ( 2 + h ) − y (2) y ( 2.5 ) − y (2) vavg . = vavg . = (2 + h) − 2 ( 2.5) − 2 or 0 − 16 = = −32ft/s = −24 − 16h, if h 0 .5 Warm-up f ( x + h) − f ( x ) m= h CSC Worthy Section 2.1:5 If a ball is thrown into the air with a velocity of 40ft/s, the height in feet t seconds later is given by y = 40t – 16t2. a) Find the average velocity for the time period beginning when t = 2 and lasting . . . (i). O.5 seconds (ii). 0.1 seconds −32ft/s −25.6ft/s Is the ball rising or falling? Warm-up f ( x + h) − f ( x ) m= h CSC Worthy Section 2.1:5 If a ball is thrown into the air with a velocity of 40ft/s, the height in feet t seconds later is given by y = 40t – 16t2. a) Find the average velocity for the time period beginning when t = 2 and lasting . . . (i). O.5 seconds (ii). 0.1 seconds −32ft/s −25.6ft/s (iii). 0.05 seconds (iv). 0.01 seconds −24.2ft/s −24.8ft/s b) Estimate the instantaneous velocity when t = 2. The instantaneous velocity when t = 2 is 24ft/s. CSC Worthy Warm-up Fill in the table and use it to estimate . . . ln x − ln 4 lim x →4 x−4 Record decimals to 6 places. .251576 .250313 .250031 .25 .249967 .249688 .248450 Warm-up Use the graph below to identify all points at which f is not continuous and state the type of discontinuity. For each of these points, then determine if it is continuous from the left, the right, or neither. x= Type of Discontinuity Continuous from . . . -5 removable neither -3 jump left 1 removable neither 5 infinite neither −5 −3 1 5 MA121 Calculus 2.6: Limits at Infinity (Horizontal Asymptotes) Dr. Soltys + Limits at Infinity Precise Definition of a Limit at Infinity Let f be a function defined on some interval (a, ). Then . . . lim f ( x ) = L means that for every 0 there is a x → corresponding number N such that if x N then f x − L . ( ) Intuitively, this means that the values of f(x) can be made arbitrarily close to L by requiring x to be sufficiently large. + Limits at Infinity Precise Definition of a Limit at Infinity lim f ( x ) = L if xN x → then f ( x) − L . − Limits at Infinity Precise Definition of a Limit at Infinity Let f be a function defined on some interval (- ,a). Then . . . lim f ( x ) = L means that for every 0 there is a x →− corresponding number N such that if x N then f x − L . ( ) Intuitively, this means that the values of f(x) can be made arbitrarily close to L by requiring x to be sufficiently small. − Limits at Infinity Precise Definition of a Limit at Infinity lim f ( x ) = L if xN x →− then f ( x) − L . Limits at Infinity Horizontal Asymptotes The line y = L is called a horizontal asymptote of the curve y = f(x) iff . . . lim f ( x ) = L x → or lim f ( x ) = L x →− Horizontal asymptotes are always of the form y=L Limits at Infinity Vertical Asymptotes Horizontal Asymptotes lim f ( x) = lim f ( x) = b x→a Equation: x = a x=3 lim f ( x) = − x →3− lim f ( x) = x →3+ x → Equation: x = b Limits at Infinity Vertical Asymptotes Horizontal Asymptotes lim f ( x) = lim f ( x) = b x→a Equation: x = a lim f ( x) = − x →0 − x=0 lim f ( x) = − x →0 + x → Equation: x = b Limits at Infinity Vertical Asymptotes Horizontal Asymptotes lim f ( x) = lim f ( x) = b x → x→a Equation: x = a Equation: y = b lim f ( x) = 1 x →− lim f ( x) = − x →0 − x=0 lim f ( x) = − x →0 + 2 lim f ( x) = 1 x → y= 1 2 2 Limits at Infinity Here are some common functions with limits at infinity that you should be able to quickly recall and sketch. Limits at Infinity 1. Exponential and Logarithmic Functions y=0 x lim e =0 x →− −x lim 2 = x →− x lim e = x → −x lim 2 =0 x → y=0 lim(− log 2 x) = lim ln x = − x →0 + x=0 lim ln x = x → x →0 + x=0 lim( − log 2 x) = x → Limits at Infinity 2. Arctangent y = −1 lim tan x = x → 2 2 y = − −1 lim tan x = − x →− 2 2 Limits at Infinity 3. Graphs of y=0 1 x Horizontal Asymptotes? Vertical Asymptotes? n lim 1 = 0 x →− x 1 lim x →− x2 1 =0 lim x → x 1 lim x → x2 y=0 x=0 =0 x=0 y=0 1 lim x →− lim 1 x → x=0 =0 x 3 x3 =0 =0 1 =0 lim − x →− x lim − 1 =0 x → x y=0 x=0 Limits at Infinity 4. Polynomial Functions of Degree n Even-degree Polynomial Odd-degree Polynomial + leading coefficient lim f ( x) = lim f ( x) = − x →− x →− lim f ( x) = x → lim f ( x) = x → End Behavior: What happens to f(x) as x approaches positive and negative infinity? Limits at Infinity 4. Polynomial Functions of Degree n Even-degree Polynomial f ( x) = x 4 − 2 x 3 + 2 x 2 − 1 Odd-degree Polynomial What if the leading coefficient is - ? f ( x) = .06 x 3 + .05 x + 1 f ( x) = x 4 − 2 x 2 − 1 ff((xx))==−xx3 3−+33xx2 2+−11 f (f x()x= ) =− x 4 − + 2 x3 + − 2x2 + −1 f ( x) = x 5 − 4 x 4 − 3x 3 − x 2 − 3x − 5 lim f ( x) = x → lim f ( x) = − and lim f ( x) = x → x →− How to Calculate Limits at Infinity for Rational Functions Horizontal Asymptotes 7 x + 16 x + 2 x How do I find lim ? 3 x→ 13 x − 2 x − 32 3 2 Horizontal asymptotes can be found by finding the limit of the quotient when the numerator and denominator are divided by the greatest power of x as x approaches If the rational function is of the form polynomial/polynomial, then . . . P( x) P( x) lim x →− Q( x) = lim x → Q( x) and there is only one horizontal asymptote. 7 x + 16 x + 2 x Find the horizontal asymptote for f ( x) = 13 x 3 − 2 x − 32 3 To find the limit as x Divide the numerator and the denominator by the largest power of x ... 3 2 7 x 16 x 2 x + 3 + 3 3 x f ( x) = x 3 x 13x 2 x 32 − 3− 3 3 x x x 2 Verify with table. 16 2 + 2 x x Simplify f ( x) = Therefore, y = 713 2 32 13 − 2 − 3 is a horizontal x x asymptote for f(x) 16 2 7+ + 2 7+0+0 x x = Find the limit of f(x) as lim 2 32 x → 13 − 2 − 3 13 − 0 − 0 x approaches x x 7+ How to Calculate Limits at Infinity for Do you notice any Rational Functions pattern? When will Horizontal Asymptotes we get y = 4x3 + 2x 2 + 5 f ( x) = 3 8 x − 7 x − 12 Horizontal: y = 1 2 Is this possible? 6 x 2 − 13x + 8 g ( x) = 5 x 2 + 11x Horizontal: y = 6 12 x 3 − x 2 + x − 23 h( x ) = 21x 4 − 15 x 2 + 2 x Horizontal: y = 0 x 3 + 8 x − 13 f ( x) = 2 9 x + 3x − 54 5 no horizontal asymptotes This graph appears to be asymptotic to a “diagonal” line. This line is a slant or oblique asymptote. 0 or ? How to Calculate Limits at Infinity for Rational Functions Horizontal Asymptotes Horizontal Asymptote “Shortcut” If the degree of the numerator and the denominator are equal, then the horizontal asymptote is the ratio of the leading coefficients If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0 If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote How to Calculate Limits at Infinity for Do you notice any Rational Functions pattern? When will Horizontal Asymptotes Verify the shortcuts. 4x3 + 2x 2 + 5 f ( x) = 3 8 x − 7 x − 12 Horizontal: y = 1 6 x 2 − 13x + 8 g ( x) = 5 x 2 + 11x Horizontal: y = 6 12 x 3 − x 2 + x − 23 h( x ) = 21x 4 − 15 x 2 + 2 x Horizontal: y = 0 x 3 + 8 x − 13 f ( x) = 2 9 x + 3x − 54 we get y = 2 5 no horizontal asymptotes 0 or ? 9x + 6 Find the horizontal asymptote for f ( x) = 5x − 1 2 To find the limit as x + . . . Divide the numerator and the denominator by the largest power of x What is does this equal as we approach + ? x2 = x Simplify Find the limit of f(x) as x approaches x = x2 9 x2 6 + 2 2 x f ( x) = x 5x 1 − x2 x2 6 9+ 2 x f ( x) = 5x 1 − x x 3 9+0 = lim x→ 5 − 0 5 Therefore, y = 3 5 is a horizontal asymptote for f(x) 9x + 6 Find the horizontal asymptote for f ( x) = 5x − 1 2 To find the limit as x - . . . Divide the numerator and the denominator by the largest power of x What is does this equal as we approach - ? x2 = x Simplify Find the limit of f(x) as x approaches 9 x2 6 + 2 2 x f ( x) = x 5x 1 − x2 x2 6 9+ 2 x f ( x) = 5x 1 − −x −x 3 9+0 = − lim x→ −5 + 0 5 y=3 5 y=3 5 Therefore, y = − 3 5 is a horizontal asymptote for f(x) Limits at Infinity Horizontal Asymptotes A function may have two horizontal asymptotes if the function has a radical variable expression or an absolute value quantity. Arctan also has two horizontal asymptotes. It is incorrect to think that lim ( x − x ) = − = 0 2 x → Infinity is not a number and the limit laws do not apply in this situation. How can we find this limit? Limits at Infinity Method 1 lim ( x − x ) = lim x ( x − 1) 2 x → x → = lim x lim ( x − 1) = = x → x → Method 2 Consider the end behavior of a polynomial function with degree 2. Limits at Infinity Vertical Asymptotes 1. If possible, factor the numerator and denominator. 2. Cancel any common factors (holes). 3. Any remaining domain restrictions are vertical asymptotes. Set equal to 0 and solve for x. 4. State the equations of all vertical asymptotes as x = a. Horizontal Asymptotes 1. Find the limit of f(x) as x approaches positive and negative infinity. 2. State the equations of all horizontal asymptotes as y = b. Practice Find all of the vertical and horizontal asymptotes for . . . Hole: ( −2, 213) x + 2x y= x − 9 x − 22 2 2 x + 2x x lim = lim = x − 9 x − 22 x − 11 x + 2x x lim = lim = − x − 9 x − 22 x − 11 x + 2x lim =1 x − 9 x − 22 x + 2x lim =1 x − 9 x − 22 2 x →11+ x → 2 2 Vertical : Asymptote Horizontal : Asymptote x = 11 x →11− x → 2 2 y =1 x → + 2 2 x → − 2 Practice Find all of the vertical and horizontal asymptotes for . . . Does this make sense? Vertical : Asymptote x = −4 x=4 Horizontal : Asymptote y=0 2 x + 10 y= x − 16 2 2 x + 10 lim = x − 16 x →−4− 2 2 x + 10 lim = − x − 16 x →−4+ 2 2 x + 10 2 x + 10 lim = − lim = x − 16 x − 16 x →4− 2 2 x + 10 lim =0 x − 16 x →− 2 x →4+ 2 2 x + 10 lim =0 x − 16 x → 2