《微积分》课程教学资源（PPT讲稿）Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number x-yayr (r=L m number

f x L x a = → lim ( )  number   Does not exist number   + − Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: Consider the function f(x)= Domain=R-o x Study the behavior of the function around X=0. Find 0 x→0+x x 0.1 10 0.01 100 0.001 1000 0.0001 10000 Find X→ x Vertical Asymptote 0.001100000

Example: x 1 Find : lim x→0+ Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs x f x 1 ( ) = Consider the function: Study the behavior of the function around x=0. x 1 Find : lim x→0− Vertical Asymptote Domain = R −{0} 0.1 10 0.01 100 0.001 1000 0.0001 10000 0.000001 1000000 x x 1 = +

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: lim nonzero Find lin x→1+ lin x-a Find lin x少1-x-1 l-x-a X x 10 x 1.01 100 x 1.001 1000 0.9 -10 Vertical Asymptote 1.0001 10000 0.999 1000 1.0000011000000 0.99999 10000 Domain=R-19 0.999999-100000

Example: 1 1 Find : lim x→1+ x − Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs 1 1 Find : lim x→1− x − Vertical Asymptote Domain = R −{1} 1.1 10 1.01 100 1.001 1000 1.0001 10000 1.000001 1000000 x 1 1 x − 0.9 -10 0.999 -1000 0.99999 -10000 0.999999 -1000000 x 1 1 x − 0 lim = nonzero= + x→a+ x − a 1 lim = − x→a− x − a 1 lim

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs lim nonzero m =+00 m x-a X-a E Xam ol e 2x 21 p EXAMPLE 9 Find lim and lim x-3 3-x-3 a) lim )IIm x→+3+x-3 x→3-x-3 )lim lin x x→4-x-4 Example x=3 a)In c)lm x→+3+3-x b)lim 4-x →4-4-x

0 = nonzero lim 3 1 ) lim →3+ x − a x 4 1 ) lim →4+ x − b x 4 1 ) lim →4− x − d x 3 1 ) lim →3− x − c x Example: x a x→ + 3− 1 ) lim 3 x b x→ + 4 − 1 ) lim 4 x d x→ − 4 − 1 ) lim 4 x c x→ − 3− 1 ) lim 3 Example: = + x→a+ x − a 1 lim = − x→a− x − a 1 lim Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs EXAMPLE 8 Find lim- if it exists x→0 1 0.5 +00 x→>0+x 0.2 0.1 100 0.05 400 0.5 lim 0.01 10.000 0.2 25 x 0.001 1.000.000 0.1 100 0.05 400 0.01 10.000 0.001 1,000.000 +00 x→0x y=2

= + → + 2 0 1 lim x x = + → − 2 0 1 lim x x = + → 2 0 1 lim x x Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs number E xample Infinite Limits Find the limits in Exercises 37-48 2x 2x2+1 x →+3-x-3 44. lim 0x(x+1) E xample 45. a. lim 2 b. lim 2 →03 x→0 a)lm b)lin x→0 x→ x x 5x+2 2 a)lm 3(x-3) 2-5x lim x

0 = number lim 3 2 ) lim →3+ x − x a x 3 2 1 ) lim 2 3 − + → − x x c x Example: 2 0 1 ) lim x a x→ Example: 2 3 ( 3) 1 ) lim → x − b x 2 3 ( 3) 1 3 ) lim − − → x x c x 5 3 ( 3) 5 2 ) lim − + → x x d x 6 3 ( 3) 2 5 ) lim − − → x x d x Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Infinite limits (a)limf(x)=+∞ (b)lim f(x)=+oo r-a+ 0 0 (c)lim flx)=-o0 (d)lim f(x) r.d

Infinite Limits + + −  −  Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs y I. The graph of f is given (a) Find each limit, or explain why it does not exist (ii) lim f(x) (v)lim x→-3 x→0 (ii) lim f(r) (iv) lim f(r) (vi) lim f(r) x-)

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote lim f(x)=too x→)a DEFINITION A line x= a is a vertical asymptote of the graph of a function y=f(r)if either imf(x)=±∞ or lim f(x)=±∞ x→a y (a)limf(x)=∞ (b)limf(x)=∞ (d)lim f(r)=-oo c) lim f(x)=-∝

Vertical Asymptote & =  → lim f (x) x a Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Find all vertical asymptotes

Find all vertical asymptotes Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs