2.3 Algebraic approach to limits

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1 CHAPTER 2. LIMITS Algebraic approac to its Now we start to learn ow to find its algebraically. Tis starts wit te simplest possible its, and ten builds tese up to more complicated examples. Fact. If C is a constant, ten C = C. Fact. x = a Fact. If f(x) and g(x) are any functions wit f(x) and g(x) bot existing ten we ave. [f(x) ± g(x)] = f(x) ± g(x) 2. [f(x)g(x)] = f(x) g(x) f(x) f(x) 3. g(x) = (as long as te it on te bottom is not zero). g(x) Comments. Using tese facts we can already do a simple it. Example. Find x 4 (3x 2 +2x + 5), algebraically, using te it laws, sowing all possible steps. x 4 (3x2 +2x + 5) Next use Fact 3. 3x +2x +5 x 4 x 4 x 4 now use use Fact x x +2 x +5 x 4 x 4 x 4 x 4 x 4 x 4 now use Facts and 2 = = 6 Comments. You can generalize te same argument as in, to any polynomial. Tat s wat te next result means. Teorem (Plug it in polynomials Teorem). If p(x) is any polynomial ten p(x) =p(a). In oter words, to find te it, JUST PLUG IT IN! Teorem 2. If f(x) is any combination (i.e. sum, product, fraction, composition, etc.) of basic functions (i.e. powers of x, exponentials, trig functions, inverses, etc.) and x = a is in te domain of f(x), ten f(x) =f(a) In oter words, to find te it, JUST PLUG IT IN! (as long as it s defined). 5x 2 +2x Example 2. Find x 3 sin π 2 x. Using Teorem we just plug in x = 3 and get sin(3π/2) = 5.

2 CHAPTER 2. LIMITS 33 Tis is were we ended on Friday, September 3 If we can t just plug it in ten we use te following result. Fact. If (x) =f(x) for all x except x = a and (x) =L, ten f(x) =L. Comments. We can prove te previous fact later using te Squeeze Teorem. But for now, it sould be pretty believable anyway: f(x) and (x) are te same except at x = a. So, if we talk about te it of f(x), wic involves only x near a, ten we will get te same result as talking about te it of (x). x 2 + x 6 Example 3. Find. x 2 x 2 x 2 + x 6 x 2 x 2 (x 2)(x + 3) x 2 x 2 x + 3 x 2 =2+3=5 now factor now cancel x 2 now just plug in Te crucial teoretical step is were we cancel x 2. Note tat te functions x 2 and x2 + x 6 are not exactly te same function. Tey are te same everywere x 2 except x = 2, were te first is defined and te second is not. By te Corollary of te Squeeze Teorem tese two functions do ave te same it. x 2 +5x +4 Example 4. Find x 4 x 2 +3x 4. x 2 +5x +4 x 4 x 2 +3x 4 (x + )(x + 4) = = x 4 (x + 4)(x ) = x 4 x + x = 4+ 4 = 3 5 = 3 5 now factor now cancel te factor x +4 now just plug in Again, te crucial teoretical step is tat it s OK to cancel. Here are some rules of tumb for algebraic manipulation of its: If te it of f(x) involves division by 0, ten factor f and/or g, cancel te g(x) factor tat gives te 0. If te it of f(x) g(x) (x) involves division by 0, ten combine te fractions p(x) using a greatest common denominator, ten factor tings and cancel te part giving division by 0.

3 CHAPTER 2. LIMITS 34 f(x) g(x) If te it of involves division by 0, ten rationalize te (x) numerator: f(x) g(x) f(x)+ g(x) f(x) g(x) = (x) f(x)+ g(x) (x)( f(x)+ g(x)) ten factor tings and cancel te part giving division by 0 Example 5. Find ( + ) 0 ( + + ) 0 ( + + ) = +0+ = rationalize te numerator finis rationalizing cancel and cancel te Just plug it in Example 6. Find t 0 t t 0 t Tis is were we ended on Monday, September 6 t + t 0 t t + t + t 0 t + t 0 t 0 t 0 t t t(t + ) t 0 t + Get a common denominator finis getting common denominator combine fractions cancel and factor bottom cancel t = = Just plug in 0+

4 CHAPTER 2. LIMITS 35 Teorem 3 (Squeeze Teorem). Let g(x) f(x) (x) for all x near x = a (except possibly at x = a). Suppose Ten g(x) (x) =L. f(x) =L Proof. We ave tat te y-values of g(x), and te y-values of (x) are becoming infinitely close to L. Butsincef(x) isbetweeng(x) and (x), te y-values of f(x) must also become infinitely close to L. Example 7. Make up graps tat illustrate te Squeeze Teorem. We start by making up two graps tat will be te top and bottom, but tat squeeze togeter at one point. Ten, between tese curves, we try to draw any curve we feel like, but wit te requirement tat it stay between te top and bottom. Te curve in te middle will ave no coice but to go troug te squeeze point. Example 8. Find x 0 x 2 sin(/x) + using te Squeeze Teorem and grap te results. Recall tat sin(), were represents anyting. Tus x 2 sin(/x)+ x 2 () + were we replaced sin(/x) wit. Similarly, sin() and so Combining tese inequalities we ave x 2 ( ) + x 2 sin(/x)+. x 2 ( ) + x 2 sin(/x)+ g f x 2 () + Now we are alf-way to using te Squeeze Teorem, but we need to verify tat te its of g and are equal. Here we can use te JUST PLUG IT IN results: g(x) = 0( ) + = (x) = 0() + = x 0 x 0

5 CHAPTER 2. LIMITS 36 Since tese are equal, we ave Here is te grap: terefore, by Sq. Tm, x 0 f(x) = Example 9. (Stewart 6t ed, Section 2.3#38) Find x 0 + xe sin(π/x). Since sin( ) (were we could ave anyting inside of sine) we ave e sin(π/x) e and so xe sin(π/x) xe. Similarly, sin( ), we ave e e sin(π/x) and xe xe sin(π/x). Combining tese statements we get xe xe sin(π/x) xe. We ll apply te Squeeze Teorem Now we ceck te conditions xe g(x) xe sin(π/x) f(x) xe. (x) g(x) = xe = 0e =0 x 0 + x 0 + (x) = xe= 0e =0 x 0 + x 0 + 0=0 Terefore tese its are te same. Terefore, by te Squeeze Teorem, we ave f(x) = 0. x 0 + Tis example also as a nice picture tat illustrates te Squeeze Teorem.

6 CHAPTER 2. LIMITS 37 Example 0. Prove te following major result: = x 0 x Suppose first tat te angle θ is between 0 and π/2. If we draw θ in a unit circle, te s pictured below can be compared as sown Labelling tese quantities we ave sin(θ) θ tan(θ) cos(θ) A = 2 cos(θ)sin(θ) A = θ 2π π 2 = θ 2 We arrange tese by size and calculate A= 2 tan(θ) 2 cos(θ)sin(θ) θ 2 cos(θ)sin(θ) θ 2 tan(θ) sin(θ) cos(θ) θ cos(θ) sin(θ) cos(θ) cos(θ) sin(θ) cos(θ) θ now mult by 2, and replace tan(θ) now divide by sin(θ) now take inverses

7 CHAPTER 2. LIMITS 38 Note tat cos(x) = cos(x) =. x 0 + Terefore, by te Squeeze Teorem, x 0 + = x 0 + x Finally, since is an odd function, and x is an odd function, we ave tat x is an even function. Terefore equals te same ting as x 0 x x 0 + x. Since bot te left and rigt and its equal, te two sided it equals as well. Tis is were we ended on Tuesday, September 7.