Curve Sketching Limits with Infinity Asymptotes

1
Curve SketchingLimits with InfinityAsymptotes

• Lesson 4.4

2
What Happens?

• We wish to investigate what happens when
  functions go

3
Limits with Infinity

• What happens to a function in the long run

N1
4
Rules for Manipulating Limits

• Note rules on page 218
• Note special limits

n is a positive rational number
k gt 0
5
Manipulating, Evaluating

• Symbolically
• Use Calculatorlimit((x2)/((3x-5),x,?)
• Graph and observe

6
Rational Functions

• Leading terms dominate
• m n gt limit an/bm
• m gt n gt limit 0
• m lt n gt asymptote linear diagonal
  or higher power polynomial

7
Rational Functions

• Vertical asymptotes
• where denominator 0
• Y-intercepts
• where x 0
• X-intercepts
• where numerator 0

8
Example

• Find
• horizontal asymptote
• vertical asymptote(s)
• zeros
• y-intercept

9
Example

• Find
• horizontal asymptote
• vertical asymptote(s)
• zeros
• y-intercept

10
Finding Other Asymptotes

• Use PropFrac to get
• If power of numerator is larger by two
• result of PropFrac is quadratic
• asymptote is a parabola

11
Example

• Consider
• Propfrac gives

12
Example

• Note the parabolic asymptote

13
Vertical Tangents

• Consider a function continuous at P(c, f(c))
• Vertical tangent at P ifare either both

c
14
Cusp

• Again, consider a function continuous at P(c,
  f(c))
• A cusp exists whenare both infinite and
  opposite in sign

15
Assignment

• Lesson 4.4
• Page 227
• Exercises 5 47 odd