Section 2.3: Calculating Limits using the Limit Laws

Transcription

1 Section 2.3: Calculating Limits using te Limit Laws In previous sections, we used graps and numerics to approimate te value of a it if it eists. Te problem wit tis owever is tat it does not always give us te correct answer, it may only provide and approimate it, or even worse, it may suggest a it eists wen in fact it doesn t. Terefore, we need some rules to elp evaluate its of certain common functions and ow to evaluate its under certain algebraic operations.. Te Basic Limit Laws We start by listing te basic it laws. Result.. Suppose tat f() and g() eist and c is a constant and n is a positive integer. Ten te following are true: (i) Sum Law: (f() + g()) = f() + g() Difference Law: (f() g()) = f() g() (iii) Constant Multiple Rule : (iv) Product Law: (v) Quotient Law: (vi) Power Law: (vii) Root Law: cf() = c f() (f() g()) = f() g() ( ) f() = f() g() g() (f())n = ( f()) n n f() = n f() provided if n is even we ave f() > 0. In addition to tese it laws, we sall assume te following its of two simple functions:

2 2 Result.2. (Limits of Constant and Identity Functions) If c is a constant, te following its are true: (i) c = c = a Te it laws can be used to evaluate its for many algebraic functions. We illustrate wit some eamples. Eample.3. Evaluate te following its stating te it laws used in eac step. (i) Quotient 2 = 2 2 Difference 2 = Power = ( 2 ) Constant Multiple = ( 2 ) Constant = ( 2 ) 2 Identity 2 2 = ( 2 ) 2 = 0 8 e2 + Constant 8 e2 + = e 2 + Eample.4. Sow tat te it of a difference of functions may eist even toug te individual its eist. Consider te functions f() = and g() =. Ten we ave f() g() = 0 for 0, so te it eists at = 0 and is equal to 0. However, te its of te functions f() and g() do not eist. Note tat tis tells us tat in order to apply te it laws, te it of te functions must eist - owever, just because te its do not eist, does not mean tat te it of te corresponding combination of functions does not eist. Our previous results and te previous eamples suggest te following result.

3 Result.5. (Direct Substitution) If f is an algebraic function and a is in te domain of f(), ten f() = f(a). In particular, its of all rational functions and polynomials can be evaluated using direct subsitution. 2. Oter Limit Laws Te net eample sows a similar result to direct substitution olds in certain special circumstances even toug a function may not be defined at a particular point. Eample 2.. Evaluate Observe tat ( 2)( + 3) = = for all 2. In particular, if tey agree for all different values ecept at = 2, ten tey must ave te same values close to = 2 and consequently te same it. Tus by direct subsitution. ( 2)( + 3) = = + 3 = Note tat in order to find te it in te previous eample, we used te fact tat te function agreed wit anoter function at all points ecept te point were we were evaluating te it and noting tat te it does not care about wat appens at te point. Tis can be formalized as follows: Result 2.2. (Replacement Teorem) If f() = g() for all in an open interval containing = a ecept possible wen = a, ten f() = g() provided tis it eists. We illustrate wit a couple of oter eamples. Eample 2.3. We ave ( + ) 2 2 (i) Evaluate ( + ) 2 2. = = = 2+ = 2

4 4 by te replacement Teorem and direct substitution. We ave t 0 t t 2 + t t 0 t t 2 + t = t 0 t 2 + t t 3 + t 2 t t 3 + t = 2 t 0 t 2 t 2 (t + ) = t 0 by te replacement Teorem and direct substitution. t + = In addition to te laws we ave already stated, tere are a number of oter results wic can be used to calculate its. We list tem below and ten finis wit some eamples of ow tey can be applied. Result 2.4. If f() g() wen is near a (ecept possibly at a) and ten its of bot f and g eist at a, ten f() g(). Result 2.5. (Te Squeeze Teorem) If f() g() () wen is near a (ecept possibly at a) and ten its of bot and f eist at a and f() = () = L ten Eample 2.6. does not eist. Observe tat and (i) Sow tat g() = L = 0 =. Since te left and rigt and its do not matc, it follows tat te it does not eist at =. Prove tat 0 2 cos = 0. Since cos (/) for all, it follows tat 2 2 cos 2.

5 5 Net observe tat since 0 2 = 2 = 0 0 by te squeeze teorem, it follows tat 0 2 cos = 0.