Curve Sketching Limits with Infinity Asymptotes

Curve SketchingLimits with InfinityAsymptotes Lesson 4.4

To infinity and beyond … What Happens? • We wish to investigate what happens when functions go …

Limits with Infinity • What happens to a function in the long run N1

Rules for Manipulating Limits • Note rules on page 218 • Note special limits n is a positive rational number k > 0

go to zero Manipulating, Evaluating • Symbolically • Use Calculatorlimit((x+2)/((3x-5),x,+) • Graph and observe

Rational Functions • Leading terms dominate • m = n => limit = an/bm • m > n => limit = 0 • m < n => asymptote linear diagonal or higher power polynomial

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

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

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

Example • Consider • Propfrac gives

Example • Note the parabolic asymptote

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

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

Assignment • Lesson 4.4 • Page 227 • Exercises 5 – 47 odd

Curve Sketching Limits with Infinity Asymptotes