2.2. Calculating Limits Using the Limit Laws. 84 Chapter 2: Limits and Continuity. The Limit Laws. THEOREM 1 Limit Laws

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1 84 Chapter : Limits and Continuit. HISTORICAL ESSAY* Limits Calculating Limits Using the Limit Laws In Section. we used graphs and calculators to guess the values of its. This section presents theorems for calculating its. The first three let us build on the results of Eample 8 in the preceding section to find its of polnomials, rational functions, and powers. The fourth and fifth prepare for calculations later in the tet. The Limit Laws The net theorem tells how to calculate its of functions that are arithmetic combinations of functions whose its we alread know. THEOREM Limit Laws If L, M, c and k are real numbers and ƒsd L and gsd M, then :c :c. Sum Rule: sƒsd + gsdd L + M :c The it of the sum of two functions is the sum of their its.. Difference Rule: sƒsd - gsdd L - M :c The it of the difference of two functions is the difference of their its. 3. Product Rule: sƒsd # gsdd L # M :c The it of a product of two functions is the product of their its. To learn more about the historical figures and the development of the major elements and topics of calculus, visit

2 . Calculating Limits Using the Limit Laws Constant Multiple Rule: sk # ƒsdd k # L :c The it of a constant times a function is the constant times the it of the function. 5. Quotient Rule: The it of a quotient of two functions is the quotient of their its, provided the it of the denominator is not zero. 6. Power Rule: If r and s are integers with no common factor and s Z 0, then provided that is a real number. (If s is even, we assume that L 7 0. ) The it of a rational power of a function is that power of the it of the function, provided the latter is a real number. L r>s :c :c sƒsddr>s L r>s ƒsd gsd L M, M Z 0 It is eas to convince ourselves that the properties in Theorem are true (although these intuitive arguments do not constitute proofs). If is sufficientl close to c, then ƒ() is close to L and g() is close to M, from our informal definition of a it. It is then reasonable that ƒsd + gsd is close to L + M; ƒsd - gsd is close to L - M; ƒ()g() is close to LM; kƒ() is close to kl; and that ƒ()>g() is close to L>M if M is not zero. We prove the Sum Rule in Section.3, based on a precise definition of it. Rules 5 are proved in Appendi. Rule 6 is proved in more advanced tets. Here are some eamples of how Theorem can be used to find its of polnomial and rational functions. EXAMPLE Using the Limit Laws Use the observations :c k k and :c c (Eample 8 in Section.) and the properties of its to find the following its. (a) (b) (c) : :c s d :c + 5 Solution (a) (b) :c s d :c :c :c :c + 5 c 3 + 4c - 3 :c s4 + - d :c s + 5d :c :c :c :c + 5 :c c4 + c - c + 5 Sum and Difference Rules Product and Multiple Rules Quotient Rule Sum and Difference Rules Power or Product Rule

3 86 Chapter : Limits and Continuit (c) : : - s4-3d Power Rule with r>s : : - 4s -d - 3 Difference Rule Product and Multiple Rules Two consequences of Theorem further simplif the task of calculating its of polnomials and rational functions. To evaluate the it of a polnomial function as approaches c, merel substitute c for in the formula for the function. To evaluate the it of a rational function as approaches a point c at which the denominator is not zero, substitute c for in the formula for the function. (See Eamples a and b.) THEOREM Limits of Polnomials Can Be Found b Substitution If Psd a n n + a n - n - + Á + a 0, then Psd Pscd a n :c cn + a n - c n - + Á + a 0. THEOREM 3 Limits of Rational Functions Can Be Found b Substitution If the Limit of the Denominator Is Not Zero If P() and Q() are polnomials and Qscd Z 0, then :c Psd Qsd Pscd Qscd. EXAMPLE Limit of a Rational Function : s -d3 + 4s -d - 3 s -d This result is similar to the second it in Eample with c -, now done in one step. Identifing Common Factors It can be shown that if Q() is a polnomial and Qscd 0, then s - cd is a factor of Q(). Thus, if the numerator and denominator of a rational function of are both zero at c, the have s - cd as a common factor. Einating Zero Denominators Algebraicall Theorem 3 applies onl if the denominator of the rational function is not zero at the it point c. If the denominator is zero, canceling common factors in the numerator and denominator ma reduce the fraction to one whose denominator is no longer zero at c. If this happens, we can find the it b substitution in the simplified fraction. EXAMPLE 3 Evaluate Canceling a Common Factor :

4 . Calculating Limits Using the Limit Laws 87 3 (, 3) Solution We cannot substitute because it makes the denominator zero. We test the numerator to see if it, too, is zero at. It is, so it has a factor of s - d in common with the denominator. Canceling the s - d s gives a simpler fraction with the same values as the original for Z : s - ds + d s - d +, if Z. Using the simpler fraction, we find the it of these values as : b substitution: (a) 3 (, 3) See Figure.8. EXAMPLE 4 Evaluate + - : - + : + Creating and Canceling a Common Factor :0. (b) FIGURE.8 The graph of ƒsd s + - d>s - d in part (a) is the same as the graph of gsd s + d> in part (b) ecept at, where ƒ is undefined. The functions have the same it as : (Eample 3). Solution This is the it we considered in Eample 0 of the preceding section. We cannot substitute 0, and the numerator and denominator have no obvious common factors. We can create a common factor b multipling both numerator and denominator b the epression (obtained b changing the sign after the square root). The preinar algebra rationalizes the numerator: # A B Therefore, A B :0 : Common factor Cancel for 0 Denominator not 0 at 0; substitute This calculation provides the correct answer to the ambiguous computer results in Eample 0 of the preceding section. The Sandwich Theorem The following theorem will enable us to calculate a variet of its in subsequent chapters. It is called the Sandwich Theorem because it refers to a function ƒ whose values are

5 88 Chapter : Limits and Continuit L h f sandwiched between the values of two other functions g and h that have the same it L at a point c. Being trapped between the values of two functions that approach L, the values of ƒ must also approach L (Figure.9). You will find a proof in Appendi. g 0 c FIGURE.9 The graph of ƒ is sandwiched between the graphs of g and h. THEOREM 4 The Sandwich Theorem Suppose that gsd ƒsd hsd for all in some open interval containing c, ecept possibl at c itself. Suppose also that Then :c ƒsd L. gsd hsd L. :c :c u() The Sandwich Theorem is sometimes called the Squeeze Theorem or the Pinching Theorem. EXAMPLE 5 Appling the Sandwich Theorem Given that 0 4 FIGURE.0 An function u() whose graph lies in the region between + s >d and - s >4d has it as : 0 (Eample 5). find :0 usd, no matter how complicated u is. Solution Since the Sandwich Theorem implies that :0 usd (Figure.0). EXAMPLE 6-4 usd + for all Z 0, s - :0 s >4dd and s + s >dd, :0 More Applications of the Sandwich Theorem (a) (Figure.a). It follows from the definition of sin u that -ƒ u ƒ sin u ƒ u ƒ for all u, and since u:0 s -ƒ u ƒd u:0 ƒ u ƒ 0, we have sin u 0. u:0 sin cos 0 (b) (a) FIGURE. The Sandwich Theorem confirms that (a) u:0 sin u 0 and (b) u:0 s - cos ud 0 (Eample 6).

6 . Calculating Limits Using the Limit Laws 89 (b) (Figure.b). From the definition of cos u, 0 - cos u ƒ u ƒ for all u, and we have u:0 s - cos ud 0 or cos u. u:0 (c) For an function ƒ(), if :c ƒ ƒsd ƒ 0, then :c ƒsd 0. The argument: - ƒ ƒsd ƒ ƒsd ƒ ƒsd ƒ and - ƒ ƒsd ƒ and ƒ ƒsd ƒ have it 0 as : c. Another important propert of its is given b the net theorem. A proof is given in the net section. THEOREM 5 If ƒsd gsd for all in some open interval containing c,ecept possibl at c itself, and the its of ƒ and g both eist as approaches c, then ƒsd gsd. :c :c The assertion resulting from replacing the less than or equal to 6 inequalit in Theorem 5 is false. Figure.a -ƒ u ƒ 6 sin u 6 ƒ u ƒ, but in the it as u : 0, equalit holds. inequalit b the strict shows that for u Z 0,